LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M. James, Mathematical Institute, 24-29 St Giles,Oxford I. 4. 5. 8. 9. 10. II. 12. 13. 15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
General cohomology theory and K-theory, P.HILTON Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British Combinatorial Conference 1973, T.P.McDONOUGH & V.C.MAVRON (eds.) An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P. ROURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillations, K.E.PETERSEN Pontryagin duality and the structure of locally compact Abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F. ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A.BELLER, R.JENSEN & P.WELCH Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)
49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87.
Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds.) Commutator calculus and groups of homotopy classes, H.J.BAUES Synthetic differential geometry, A.KOCK Combinatorics, H.N.V.TEMPERLEY (ed.) Singularity theory, V.I.ARNOLD Markov processes and related problems of analysis, E.B.DYNKIN Ordered permutation groups, A.M.W.GLASS Journ£es arithmetiques 1980, J.V.ARMITAGE (ed.) Techniques of geometric topology, R.A.FENN Singularities of smooth functions and maps, J.MARTINET Applicable differential geometry, M.CRAMPIN & F.A.E.PIRANI Integrable systems, S.P.NOVIKOV et al. The core model, A.DODD Economics for mathematicians, J.W.S.CASSELS Continuous semigroups in Banach algebras, A.M.SINCLAIR Basic concepts of enriched category theory, G.M.KELLY Several complex variables and complex manifolds I, M.J.FIELD Several complex variables and complex manifolds II, M.J.FIELD Classification problems in ergodic theory, W.PARRY & S.TUNCEL Complex algebraic surfaces, A.BEAUVILLE Representation theory, I.M.GELFAND et al. Stochastic differential equations on manifolds, K.D.ELWORTHY Groups - St Andrews 1981, C.M.CAMPBELL & E.F.ROBERTSON (eds.) Commutative algebra: Durham 1981, R.Y.SHARP (ed.) Riemann surfaces: a view towards several complex variables, A.T.HUCKLEBERRY Symmetric designs: an algebraic approach, E.S.LANDER New geometric splittings of classical knots (algebraic knots), L.SIEBENMANN & F.BONAHON Linear differential operators, H.O.CORDES Isolated singular points on complete intersections, E.J.N.LOOIJENGA A primer on Riemann surfaces, A.F.BEARDON Probability, statistics and analysis, J.F.C.KINGMAN & G.E.H.REUTER (eds.) Introduction to the representation theory of compact and locally compact groups, A.ROBERT Skew fields, P.K.DRAXL Surveys in combinatorics: Invited papers for the ninth British Combinatorial Conference 1983, E.K.LLOYD (ed.) Homogeneous structures on Riemannian manifolds, F.TRICERRI & L.VANHECKE Finite group algebras and their modules, P.LANDROCK Solitons, P.G.DRAZIN Topological topics, I.M.JAMES (ed.) Surveys in set theory, A.R.D.MATHIAS (ed.)
London Mathematical Society Lecture Note Series: 86
Topological Topics Articles on algebra and topology presented to Professor P.J. Hilton in celebration of his sixtieth birthday I.M. JAMES Mathematical Institute, University of Oxford
CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney
Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia ©Cambridge University Press 1983 First published 1983 Library of Congress catalogue card number: 83-7745 British Library Cataloguing in Publication Data Topological topics - (Lecture note series/London Mathematical Society, ISSN 0076-0552; 86) 1. Topology - Addresses, essays, lectures I. James, I.M. II. Hilton, P.J. III. Series 514 QA611.15 ISBN 0 521 27581 4 Transferred to digital printing 2002
In honour of Peter Hilton on the occasion of his sixtieth birthday
CONTENTS
Publications of P.J. Hilton
1
Essay on Hilton's work in topology Guido Mislin
15
The work of Peter Hilton in algebra Urs Stammbach
31
The dual Whitehead theorems J.P. May
46
Homotopy cocomplete classes of spaces and the realization of the singular complex D. Puppe
55
Betti numbers of Hilbert modular varieties E. Thomas and A.T. Vasquez
70
Homotopy pairs in Eckmann-Hilton duality K.A. Hardie
88
Profinite Chern classes for group representations Beno Eckmann and Guido Mislin
103
Automorphisms of surfaces and class numbers: an illustration of the G-index theorem John Ewing
120
The real dimension of a vector bundle at the prime two Edgar H, Brown, Jr. and Franklin P. Peterson
128
Maps between classifying spaces, III J.F. Adams and Z. Mahmud
136
Idempotent codensity monads and the profinite completion of topological groups A. Deleanu
154
Finitary automorphisms and integral homology J- Roitberg
164
Finite group actions on Grassman manifolds Henry H. Glover and Guido Mislin
169
P U B L I C A T I O N S O F P.J.
HILTON
* Book [l]
Calculating the homotopy groups of A -polyhedra I, Quart. J. Math. Oxford (2), 1 (1950), 299-309. n
[2]
Calculating the homotopy groups of A -polyhedra II, Quart. J. Math. Oxford (2), 2 (1951), 228-240. n
[3]
Suspension theorems and the generalized Hopf invariant, Proc. London Math. Soc. (3), 1 (1951), 462-493.
[4]
The Hopf invariant and homotopy groups of spheres, Proc. Cambridge Phil. S o c , 48 (1952), 547-554.
[5*]
An introduction to homotopy theory, Cambridge University Press, (1953).
[6]
(with M.G. Barratt), On join operations in homotopy groups, Proc. London Math. Soc. (3), 1 (1953), 430-445.
[7]
(with J.H.C. Whitehead), Note on the Whitehead product, Ann. of Math., 58 (1953), 429-442.
[8]
On the Hopf invariant of a composition element, J. London Math. Soc. 29 (1954), 165-171.
[9]
A certain triple Whitehead product, Proc. Cambridge Phil. S o c , 50 (1954), 189-197.
[10]
On the homotopy groups of the union of spheres, Proc. Int. Cong. Amsterdam (1954).
[ll]
On the homotopy groups of the union of spaces, Comment. Math. Helv. 29 (1955), 59-92.
[12]
Note on the P-homomorphism in homotopy groups of spheres, Proc. Cambridge Phil. Soc. 51 (1955), 230-233.
[13]
On the homotopy groups of the union of spheres, J. London Math. S o c , 30 (1955), 154-172.
[14]
Remark on the factorization of spaces, Bull. Acad. Polon. Sci. (1) III. 3 (1955) 579-581.
[15]
On Borsuk dependence and duality, Bull. S o c Math. Belg., 7 (1955), 143-155.
[16]
(with J.F. Adams), On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330.
[17]
Note on the higher Hopf invariants, Proc Cambridge Phil. S o c 52 (1956), 750-752.
[18]
Remark on the tensor product of modules, Bull.Acad. Polon. Sci. Cl. III., 4 (1956), 325-328.
[19]
Generalizations of the Hopf invariant, Colloque de Topologie Algebrique, C.B.R.M. (1956), 9-27.
Publications of P.J.
Hilton
2
[20]
Note on quasi Lie rings, Fund. Math. 43 (1956), 203-237.
[2l]
On divisors and multiples of continuous maps, Fund. Math. 43 (1957), 358-386.
[22]
(with W. Ledermann) Homology and ringoids I, Proc. Cambridge Phil. S o c , 54 (1958), 152-167.
[23]
(with B. Eckmann) Groupes d'homotopie et dualite. Groupes absolus, C.R. Acad. Sci. Paris, 246 (1958), 2444-2447.
[24]
(with B. Eckmann) Groupes d'homotopie et dualite. Suites exactes, C.R. Acad. Sci. Paris, 246 (1958), 2555-2558.
[25]
(with B. Eckmann) Groupes d'homotopie et dualite. Coefficients, C.R. Acad. Sci. Paris, 246 (1958), 2991-2993.
[26]
(with B. Eckmann) Transgression homotopique et cohomologique, C.R. Acad. Sci. Paris, 247 (1958), 629-632.
[27]
(with W. Ledermann) Homological ringoids, Coll.Math. 6 (1958), 177-186.
[28*]
Differential calculus, Routledge and Kegan Paul, (1958).
[29]
(with B. Eckmann) Decomposition homologique d'un polyedre simplement connexe, C.R. Acad. Sci. Paris, 248 (1959), 2054-2056.
[30]
(with B. Eckmann) Homology and homotopy decomposition of continuous maps, Proc. Nat. Acad. Sci., 45 (1959), 372-375.
[31]
Homotopy theory of modules and duality, Proc. Mexico Symposium, (1958), 273-281.
[32]
(with W. Ledermann) Homology and ringoids II, Proc. Cambridge Phil. Soc. 55 (1959), 129-164.
[33]
(with T. Ganea) Decomposition of spaces in cartesian products and unions, Proc. Cambridge Phil. S o c , 55 (1959), 248-256.
[34]
(with W. Ledermann) Homology and ringoids III, Proc. Cambridge Phil. S o c , 56 (1960), 1-12.
[35]
(with W. Ledermann) On the Jordan-Holder theorem in homological monoids, P r o c London Math. S o c (3), 10 (1960), 321-334.
[36]
On an generalization of nilpotency to semi-simplicial complexes, P r o c London Math. S o c (3), 10 (1960), 604-622.
[37]
(with B. Eckmann) Operators and co-operators in homotopy theory, Math. Ann. 141 (1960), 1-21.
[38]
(with I. Berstein) Category and generalized Hopf invariants, 111. J. Math., 4 (1960), 437-451.
[39*]
Partial derivatives, Routledge and Kegan Paul, (1960).
[40*]
(with S. Wylie) Homology Theory, Cambridge Univ. Press, (1960).
[41]
(with E.H. Spanier) On the embeddability of certain complexes in Euclidean spaces, Proc. Amer. Math. S o c , 11 (1960), 523-526.
[42]
Remark on the free product of groups, Trans. Amer. Math. S o c , 96 (1960), 478-488.
Publications of P.J.
Hilton
3
[43]
(with W. Ledermann) Remark on the l.c.m. in homological ringoids, Quart. J. Math. Oxford, 11 (1960), 287-294.
[44]
(with B. Eckmann) Homotopy groups of maps and exact sequences, Comment. Math. Helv., 34 (1960), 271-304.
[45]
Note on the Jacobi identity for Whitehead products, Proc. Cambridge Phil. S o c , 57 (1961), 180-182.
[46]
Memorial tribute to J.H.C. Whitehead, Enseignement Mathematique, 7 (1961), 107-125.
[47]
On excision and principal fibrations, Comment. Math. Helv., 35 (1961), 77-84.
[48]
(with B. Eckmann) Structure maps in group theory, Fund. Math., 50 (1961), 207-221.
[49]
(with D. Rees) Natural maps of extension functors and a theorem of R.G. Swan, Proc. Cambridge Phil. S o c , 57 (1961), 489-502.
[50]
Note on free and direct products in general categories, Bull. Soc. Math. Belg., 13 (1961), 38-49.
[51]
(with B. Eckmann) Group-like structures in general categories I. Multiplications and comultiplications, Math. Ann. 145 (1962), 227-255.
[52]
Fundamental group as a functor, Bull. Soc. Math. Belg., 14 (1962), 153-177.
[53]
Note on a theorem of Stasheff, Bull.Acad. Polon. 127-131.
[54]
(with T. Ganea and F.P. Peterson) On the homotopy-commutativity of loop-spaces and suspensions, Topology, 1 (1962), 133-141.
[55]
(with B. Eckmann and T. Ganea) On means in general categories, Studies in mathematical analysis and related topics, Stanford (1962), 82-92.
[56]
(with B. Eckmann) Group-like structures in general categories II. Equalizers, limits, length, Math. Ann., 151 (1963), 150-186.
[57]
(with B. Eckmann) Group-like structures in general categories III. Primitive categories, Math. Ann., 150 (1963), 165-187.
[58]
(with S.M. Yahya) Unique divisibility in abelian groups, Acta Math., 14 (1963), 229-239.
[59]
Natural group structures in homotopy theory, Lucrarile Consfatuirii de Geometrie so Topologie, Iasi, (1962), 33-37.
[60]
Nilpotency and higher Whitehead products, Proc. Coll. Alg. Top., Aarhus (1962), 28-31.
[6l]
(with B. Eckmann) A natural transformation in homotopy theory and a theorem of G.W. Whitehead, Math. Z., 82 (1963), 115-124.
[62]
Remark on loop spaces, Proc. Amer. Math. S o c , 15 (1964), 596-600.
[63]
Nilpotency and H-spaces, Topology, 3 (1964), suppl. 2, 161-176.
Sci., 10 (1962),
Publications of P.J.
Hilton
4
[64]
(with I. Berstein) Suspensions and comultiplications, Topology, 2 (1963), 63-82.
[65]
(with I. Berstein) Homomorphisms of homotopy structures, Topology et Geom. Diff. (Sem. C. Ehresmann), Paris (1963), 1-24.
[66]
(with B. Eckmann) Unions and intersections Comment. Math. Helv., 38 (1964), 293-307.
[67]
Spectral sequences for a factorization of maps, Seattle Conference (1963).
[68]
(with B. Eckmann) Composition functors and spectral sequences, Comment. Math. Helv., 41 (1966-67), 187-221.
[69]
Note on H-spaces-and nilpotency, Bull. Acad. Polon. Sci. 11 (1963), 505-509.
[70]
(with B. Eckmann) Exact couples in abelian categories, J. Algebra, 3 (1966), 38-87.
[71]
Categories non-abeliennes, Sem. Math. Sup., Universite de Montreal (1964) .
[72]
Exact couples for iterated fibrations, Centre Beige de Rech. Math., Louvain (1966), 119-136.
[73]
Correspondences and exact squares, Proc. Conf. Cat. Alg., La Jolla, Springer (1966), 254-271.
[74]
(with B. Eckmann) Filtrations, associated graded objects and completions, Math. Z., 98 (1967), 319-354.
[75]
Review of 'Modern algebraic topology1 by D.G. Bourgin, Bull. Amer. Math. Soc. 71 (1965), 843-850.
[76]
The continuing work of 145-149.
[77*]
Homotopy theory and duality, Gordon and Breach (1965).
[78]
(with I. Pressman) A generalization of certain homological functors, Ann. Mat. Pura Appl. (4), 71 (1966), 331-350.
[79]
On the homotopy type of compact polyhedra , Fund. Math., 61 (1967), 105-109.
[80]
On systems of interlocking exact sequences, Fund. Math., 61 (1967), 111-119.
[81*]
Studies in modern topology, Introduction, Math. Ass. Amer., Prentice-Hall (1968).
[82]
The Grothendieck group of compact polyhedra, Fund. Math., 61 (1967), 199-214.
[83]
Arts and Sciences, Methuen, (1967), 20-46.
[84]
(with A. Deleanu) Some remarks on general cohomology theories, Math. Scand., 22(1968), 227-240.
[85]
Some remarks concerning the semi-ring of polyhedra, Bull. Soc. Math. Belg., 19 (1967), 277-288.
in homotopy categories,
CCSM, Arithmetic Teacher (1966),
Publications of P.J.
Hilton
5
[86]
(with B. Eckmann) Commuting limits with colimits, J. Alg., 11 (1969), 116-144.
[87]
(with B. Eckmann) Homotopical obstruction theory, An. Acad. Brasil. Cienc. 40 (1968), 407-429.
L88]
Filtrations, Cahiers Topologie Geom. Differentielle, 9 (1967), 243-253.
[89]
On the construction of cohomology theories, Rend. Mat. (6), 1 (1968), 219-232.
[90]
On commuting limits, Cahiers Topologie Geom. Differentielle, 10 (1968), 127-138.
[9l]
Categories and functors, Probe, (1968), 33-37.
[92]
Note on the homotopy type of mapping cones, Comm. Pure Appl. Math. 21 (1968), 515-519.
[93]
(with J. Roitberg) Note on principal Math. Soc. 74 (1968), 957-959.
[94J
(with I. Pressman) On completing bicartesian squares, Proc. Symp. Pure Math., Vol. XVII, Americ. Math. Soc.(1970), 37-49.
[95*]
(with H.B. Griffiths) Classical mathematics, Van Nostrand Reinhold (1970) .
[96J
(with B. Steer) On fibred categories and cohomology, Topology, 8 (1969), 243-251.
[97]
(with R. Douglas and F. Sigrist) H-spaces, Springer Lecture Notes, 92 (1969), 65-74.
[98]
(with J. Roitberg) On principal Math., 90 (1969), 91-107.
[99]
On factorization of manifolds, Proc. 15th Scand. Math. Congress, Oslo, Springer Lecture Notes 118 (1970), 48-57.
S -bundles, Bull. Amer.
S -bundles over spheres, Ann.
[lOO]
(with Y.C. Wu) On the addition of relations in an abelian category, Canad. J. Math., 22 (1970), 66-74.
[101]
On the Ditchley conference and curricular reform, Amer. Math. Monthly, 75 (1969), 1005-1006.
[102*]
Algebraic Topology, Courant Institute of Math. Sciences, NYU (1969) .
[103*]
General cohomology theory and K-theory, Lond. Math. S o c , Lecture Note Series 1, Cambridge University Press (1971).
[104]
(with A. Deleanu) On the general Cech construction of cohomology theories, Symposia Mathematica, 1st. Naz. di Alt. Mat., (1970), 193-218.
[105]
(Ed.) Proc. Battelle Conf. on category theory, homology theory and their applications, Springer Lecture Notes, Vols. 86, 92, 99 (1969) .
Publications of P.J.
Hilton
6
[106]
(with H. Hanisch und W.M. Hirsch) Algebraic and combinatorial aspects of coherent structures, Trans. N.Y. Acad. Sci. II, 31 (1969), 1024-1037.
[107]
(with R. Long and N. Meltzer) Research in mathematics education, Educational Studies in Mathematics, 2 (1970), 446-468.
[108]
(with A. Deleanu) On the generalized Cech construction of cohomology theories, Battelle Institute Report No. 28, Geneva (1969), 35.
[109]
On the category of direct systems and functors on groups, Battelle Institute Report No. 32, Geneva (1970), 40.
[110]
Putting coefficients into a cohomology theory, Battelle Institute Report No. 33, Geneva (1970), 34.
[Ill]
(with A. Deleanu) On Kan extensions of cohomology theories and Serre classes of groups, Battelle Institute Report No. 34, Geneva (1970), 40.
[112]
(with J. Roitberg) On the classification problem for of rank 2, Comment. Math. Helv., 45 (1970), 506-516.
[113]
Kategorien, Funktoren und natiirliche Transformationen, Math. Phys. Semesterberichte, 17 (1970), 1-12.
[114]
Cancellation: A Crucial Property? New York State Math. Teachers Journal, 20 (1970), 68-74, 132-135.
H-spaces
[115]
The Cosrims Reports, Amer. Math. Monthly, 77 (1970), 515-517.
[116]
Putting coefficients into a cohomology theory, Proc. Koninkl. Nederl. Akad. van Wetenschappen, Series A, 73 (1970), 196-216.
[117]
On the category of direct systems and functors on groups, Pure Appl. Alg., 1 (1971), 1-26.
[118]
(with A. Deleanu) On Kan extensions of cohomology theories and Serre classes of groups, Fund. Math., 73 (1971), 143-165.
[119]
(with J. Roitberg) Note on quasifibrations and fibre bundles, 111. J. Math., 15 (1971), 1-8.
[120]
(with G. Mislin and J. Roitberg) Sphere bundles over spheres and non-cancellation phenomena, Springer Lecture Notes 249 (1971) 34-46.
[121*]
Lectures on Homological Algebra, Reg. Conf. Series in Math., Amer. Math. Soc. (1971).
[122]
(with B. Eckmann) On central extensions and homology, Battelle Institute Report No. 45, Geneva (1971), 18.
[123]
On filtered systems of groups, colimits, and Kan extensions, J. Pure Appl. Alg., 1 (1971), 199-217.
[124]
Extensions of functors on groups and coefficients in a cohomology theory, Actes Congres intern, math., 1970, Tome 1, 1-6.
[125]
Topology in the high school, Educational Studies in Mathematics 3 (1971), 436-453.
Publications of P.J.
Hilton
7
[126]
The role of categorical language in pre-college mathematical education.
L127J
Should mathematical logic be taught formally in mathematics classes? The Mathematics Teacher, 64 (1971), 389-394.
[128]
Direkte Systeme von Gruppen, Math. Phys. Semesterberichte, 18 (1971), 174-193.
[129]
(with J. Roitberg) On the classification of torsion-free H-spaces of rank 2, Springer Lecture Notes 168 (1970), 67-74.
[130]
(with A. Deleanu) Remark on Cech extension of cohomology functors, Proc. Adv. Stu. Inst. Alg. Top., Aarhus (1970), 44-66.
[131]
(with A. Deleanu) On the splitting of universal coefficient sequences, Proc. Adv. Stu. Inst. Alg. Top., Aarhus (1970), 180-201.
[132]
On direct systems of groups, Bol. Soc. Brasil. Mat. 2 (1970), 1-20.
[133]
(with J. Roitberg) On quasifibrations and orthogonal bundles, Springer Lecture Notes 196 (1971), 100-106.
[134]
(with B. Eckmann and U. Stammbach) On Ganea's extended homology sequence and free presentations of group extensions, Battelle Institute Report No. 48, Geneva (1971), 23.
[135]
(with A. Deleanu) Localization, homology and a construction of Adams, Battelle Institute Report No. 47 (1971), 31.
[136]
(with B. Eckmann and U. Stammbach) On the homology theory of central group extensions, Comment. Math. Helv., 47 (1972), 102-122.
[137]
Some problems of contemporary university education, Bol. Soc. Brasil Mat., 2 (1971), 67-75.
[138]
Non-cancellation phenomena in topology, Coll. Math. Soc. Janos Bolyai 8. Topics in Topology, Keszthely, Hungary (1972), 405-416.
[139]
(with B. Eckmann and U. Stammbach) On the homology theory of central group extensions II. The exact sequence in the general case, Comment. Math. Helv. 47 (1972), 171-178.
[140 ]
(with B. Eckmann) On central group extensions and homology, Comment. Math. Helv. 46 (1971), 345-355.
[141* ] Topicos de algebra homologica, Publicagoes do Instituto de Matematica e Estatistica, Universidade de Sao Paulo (1970). [142* ]
(with U. Stammbach) A course in homological algebra, Springer (1971).
[143]
Topologie a l'ecole secondaire, Niko 8 (1971), 2-28; (in dutch) 69-95.
[144]
Suites spectrales et theories de cohomologie generales, Universite de Montpellier, Publication No. 88 (1969-1970).
[145]
Melanges d'algebre pure et appliquee, Universite de Montpellier, Publication No. 83 (1969-1970).
[146]
The art of mathematics, Kynoch Press, Birmingham, 1960, 14.
Publications of P.J.
Hilt/on
8
[147]
Arithmetic - down but not out
Teaching Arithmetic, (1963), 9-13.
[148]
Arithmetic as part of mathematics, Mathematics Teaching, 27 (1964).
[149]
Categorieen en functoren, Niko 9 (1971), 85-100.
[150]
(with G. Mislin and J. Roitberg) Note on a criterion of Scheerer, 111. Journ. Math. 17 (1973), 680-688.
[151]
Obituary: Heinz Hopf, Bull. London Math. Soc. 4 (1972), 202-217.
[152]
(with G. Mislin and J. Roitberg) H-Spaces of rank two and noncancellation phenomena, Invent. Math. 16 (1972), 325-334.
[153]
Categories and homological algebra, Encyclopaedia Britannica, (1974), 547-554.
[154]
Extensions of functors on groups and coefficients in a cohomology theory, Actes du Congres International des Mathematiciens, Nice 1970, Tome 1, 323-328.
[155]
(with B. Eckmann and U. Stammbach) On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra, 3 (1973), 73-82.
[156]
(with G. Mislin and J. Roitberg) Topological localization and nilpotent groups, Bull. Amer. Math. Soc. 78 (1972), 1060-1063.
[157]
(with G. Mislin and J. Roitberg) Homotopical localization, Proc. London Math. Soc. (3), 24 (1973), 693-706.
[158]
(with A. Deleanu) Localization, homology and a construction of Adams, Trans Amer. Math. Soc. 179 (1973), 349-362.
[159]
The language of categories in high school mathematics, Niko 15 (1973) , 17-55.
[l60]
A mathematician's miscellany, Battelle Seattle Research Center Lecture Series 72-1 (1972).
[l6l]
(with G. Mislin and J. Roitberg) Sphere bundles over spheres and non-cancellation phenomena. London Math. Soc. 6 (1972), 15-23.
[162]
(with U. Stammbach) Two remarks on the homology of group extensions, J. Austral. Math. S o c , 17 (1974), 345-357.
[163*]
Category theory, NSF Short Course, Colgate University (1972).
[l64]
In the bag, New York State Mathematics Teachers' Journal, 23 (1973), 8-11.
[165]
(with U. Stammbach) On torsion in the differentials of the Lyndon-Hochschild-Serre spectral sequence, J. Algebra 29 (1974), 349-367.
[166]
(with U. Stammbach) On the differentials of the Lyndon-HochschildSerre spectral sequence, Bull. Amer. Math. Soc. 79 (1973), 796-799.
[l67]
Localization and cohomology of nilpotent groups, Math. Z. 132 (1973), 263-286.
Publications of P.J.
Hilton
9
[168]
(with A. Deleanu and A. Frei) Generalized Adams completion, Cahiers Topologie GǤom. Differentielle 15 (1974), 61-82.
[169]
Remarks on the localization of nilpotent groups, Comm. Pure Appl. Math. 24 (1973), 703-713.
[170]
Some problems of contemporary education, Papers on Educational Reform, Vol. 4 (1974), Open Court Publishing Company, 77-104.
[171*]
(with Y.-C. Wu) A course in modern algebra, John Wiley and Sons, 1974.
[172]
Localization of nilpotent spaces, Springer Lecture Notes 428 (1974), 18-43.
[173]
The survival of education, Educational Technology (1973).
[174]
(with A. Deleanu and A. Frei) Idempotent triples and completion, Math. Z. 143 (1975), 91-104.
[175]
Localization in topology, Amer. Math. Monthly, 82 (1975), 113-131.
[176*]
Le langage des categories, Collection Formation des Maitres, Cedic. Paris (1973).
[177]
The category of nilpotent groups and localization, Colloque sur 1'algebre des categories, Amiens, 1973, Cahiers Topologie Geom. Differentielle 14 (1973), 31-33.
[178]
Nilpotent actions on nilpotent groups, Springer Lecture Notes, 450 (1975), 174-196.
[179]
On direct limits of nilpotent groups, Springer Lecture Notes 418 (1974), 68-77.
[180]
The training of mathematicians today, Mathematiker tiber die Mathematik, edited by Michael Otte, Springer Verlag (1974).
[181]
A case against managerial principles in education.
[182]
Ten years with Springer Verlag, Springer Verlag, New York (1974) .
[183]
Localization of nilpotent groups: homological and combinatorial methods, Comptes Rendus des Journees Mathematiques S.M.F., Montpellier, (1974), 123-132.
[184]
Education in mathematics and science today; the spread of false dichotomies, Proc. 3rd Int. Congr. Math. Ed., Karlsruhe (1976), 75-97.
[185*]
Homologie des groupes, Collection Mathematique, Universite Laval, (1973) .
[186*]
(with G. Mislin and J. Roitberg) Localization of nilpotent groups and spaces. North Holland (1975).
[187]
(with G. Mislin) Bicartesian squares of nilpotent groups, Comment. Math. Helv. 50 (1975), 477-491.
[188]
(with G. Mislin) On the genus of a nilpotent group with finite commutator subgroup, Math. Z. 146 (1976), 201-211.
Publications of P.J.
Hilton
10
[189]
(with U. Stammbach) On group actions on groups and associated series, Math. Proc. Cambridge Philos. Soc. 80 (1976), 43-55.
[190]
(with A. Deleanu) On the categorical shape of a functor, Fund. Math. 97 (1977) 157-176.
[l9l]
(with G. Mislin) Remarkable squares of homotopy types, Bol. Soc. Brasil. Mat. 5 (1974), 165-180.
[192]
(with A. Deleanu) Borsuk shape and a generalization of Grothendieck's definition of pro-category. Math. Proc. Cambridge Philos. Soc. 79 (1976), 473-482.
[193]
The new emphasis on applied mathematics, Newsletter, Conference Board of the Mathematical Sciences, vol. 10, 2 (1975), 17-19.
[194]
What is modern mathematics?, Pokroky Matematiky Fyziky a Astronomie, 3 (1977), 151-164.
[195]
(with J. Roitberg) Generalized C-theory and torsion phenomena in nilpotent spaces. Houston J. Math. 2 (1976), 525-559.
[196]
(with U. Stammbach) Localization and isolators, Houston J. Math. 2 (1976), 195-206.
[197]
A humanist's assault on managerial principles in education.
[198]
Localization of nilpotent spaces. Lecture Notes in Pure and Applied Mathematics, vol. 12, Marcel Dekker (1975), 75-100.
[199]
The anatomy of a conference.
[200]
(with J. Roitberg) On the Zeeman comparison theorem for the homology of quasi'-nilpotent fibrations, Quart. J. Math. Oxford (2), 27 (1976), 433-444.
[201]
(with A. Deleanu) On Postnikov-true families of complexes and the Adams completion, Fund. Math. 106 (1980), 53-65.
[202]
On Serre classes of nilpotent groups, Proc. Conference in Honor of Candido do Lima da Silva Dias.
[203]
Unfolding of singularities, Proceedings of the Sao Paulo Symposium on Functional Analysis, Lecture Notes in Pure and Applied Mathematics, vol. 18, Marcel Dekker (1976), 111-134.
[204]
What experiences should be provided in graduate school to prepare the college mathematics teacher?, The Bicentennial Tribute to American Mathematics, Math. Ass. Amer. (1977), 187-191.
[205]
(with A. Deleanu) Note on homology and cohomology with 2Zp-coefficients, Czechoslovak Math. J. 28 (103), 474-479.
[206]
(with A. Deleanu) On a generalization of Artin-Mazur completion (preprint).
[207]
(with R, Fisk) Derivatives without limits (preprint).
[208]
Basic mathematical skills and learning: Position Paper, NIE Conference, vol. 1 (1976), 88-92.
[209]
(with G. Rising) Thoughts on the state of mathematics education today, NIE Conference, vol. 2 (1976), 33-42.
Publications of P,J.
Hilton
11
[210]
Structural stability, catastrophe theory and their applications in the sciences, Research Futures, Battelle Institute (1976).
[21l]
Localization and cohomology of nilpotent groups, Cahiers Topologie Geom. Differentielle, 14 (1973), 341-370.
L212J
Localization in group theory and homotopy theory, Jber. Deutsch. Math.-Verein, 79 (1977), 70-78.
[213]
(with David Singer) On G-nilpotency (preprint).
[214]
(with G. Mislin and J. Roitberg) On maps of finite complexes into nilpotent spaces of finite type: a correction to 'Homotopical localization ', Proc. London Math. Soc. 34 (1978), 213-225.
[215]
On
[216]
(with A. Deleanu) Generalized shape theory, Springer Lecture Notes 609 (1977), 56-65.
[217]
Localization theories for groups and homotopy types, Springer Lecture Notes 597 (1977), 319-329.
[218]
(with G. Mislin and J. Roitberg) On co-H-spaces, Comment. Math. Helv. 53 (1978) 1-14.
[219]
Some contributions of Beno Eckmann to the development of topology and related fields, L'Enseignement Mathematique, 23 (1977), 191-207.
[220]
(with Joseph Roitberg) On the finitude of counterimages in maps of function-spaces: Correction to 'Generalized C-theory and torsion phenomena in nilpotent spaces', Houston J« Math. 3 (1977) 235-238.
[22l]
Nilpotency in group theory and topology,
[222]
A Friendship and a Bond? Semper Attentus, Beitrage fur Heinz Gotze zum 8. August 1977, Springer Verlag (1977), 150-153.
[223]
(with G. Goldhaber), NAS-NRC Committee on Applied Mathematics Training, Notices Amer. Math. Soc. (1977), 435-436.
[224]
The changing face of mathematics, Case Alumnus, 57, 2(1977), 5-7.
[225]
(with C. Cassidy) L'isolateur d'un homomorphisme de groupes, Canad. J. Math. 31 (1979), 375-391.
[226]
(with G. Mislin, J. Roitberg and R. Steiner) On free maps and free homotopies into nilpotent spaces, Springer Lecture Notes 673 (1978), 202-218.
[227]
On orbit sets for group actions and localization, Springer Lecture Notes 673 (1978), 185-201.
[228]
(with J. Roitberg) On a generalization of two exact sequences of Steiner, 111. J. Math. 24 (1980), 206-215.
[229]
Teaching and Research: A false dichotomy, Mathematical Intelligencer, 1 (1978), 76-80.
G-spaces, Bol. Soc. Brasil. Mat. 7 (1976), 65-73.
Publications of P.J.
Hilton
12
[230]
(with Joseph Roitberg and David Singer) On G-spaces, Serre classes, and G-nilpotency, Math. Proc. Cambridge Philos. Soc. 84 (1978), 443-454.
[231]
Vector spaces, abelian groups and groups: similarities and differences, Atas do 11 Coloquis da SBM, 1977.
[232]
(with Carl Bereiter and Stephen Willoughby) Real Math, Grade 1, Open Court Publishing Company (1978).
[233]
Some thoughts on math anxiety, I. Thoughts on diagnosis, Ontario Mathematics Gazette, 17, 1 (1978), 35-42.
[234]
Some thoughts on math anxiety, II. Thoughts on cure, Ontario Mathematics Gazette, 17, 2 (1978), 26-28.
[235]
(with Carl Bereiter and Stephen Willoughby) Real Math, Grade 2, Open Court Publishing Company (1978).
[236]
(with Joseph Roitberg) Profinite completion and generalizations of a theorem of Blackburn, J. Algebra 60 (1979), 289-306.
[237]
Dangerous division, California Mathematics, 4 (1979), 15-21.
[238]
(with Paulo Leite) On nilpotent spaces and C-theory, C.R. Math. Rep. Acad. Sci. Canada, 1 (1979), 125-128.
[239]
Review of 'Obstruction theory1 by Hans J. Baues, Bull. Amer. Math. Soc. 1 (1979), 292-398.
[240]
Review of 'Why the professor can't teach1 by Morris Kline, Amer. Math. Monthly, 86 (1979), 407-412.
[241]
(with Carl Bereiter and Stephen Willoughby) Real Math, Grade 3, Open Court Publishing Company (1979).
[242]
Duality in homotopy theory: a retrospective survey. J. Pure Appl. Algebra, 19 (1980), 159-169.
[243]
The role of applications in the undergraduate mathematics curriculum, Ad Hoc Committee. Applied Mathematics Training (Chairman, Peter Hilton), NRC (1979), 25.
[244]
Math anxiety; some suggested causes and cures, part I, Two-YearCollege Mathematics Journal 11 (1980), 174-188.
[245]
(with Carl Bereiter, Joseph Rubinstein and Stephen Willoughby) Real Math, Grade 4, Open Court Publishing Company (1980).
[246]
(with Jean Pedersen) Review of 'Overcoming math anxiety1 by Sheila Tobias, and 'Mind over math1 by Stanley Kogelman and Joseph Warren, Amer. Math. Monthly 87 (1980), 143-148.
[247*]
(with Jean Pedersen) Fear no more: An adult approach to mathematics, Addison Wesley (1982).
[248]
Math anxiety: Some suggested causes and cures, Part II, Two-YearCollege Mathematics Journal 11 (1980), 246-251.
[249]
(with Jean Pedersen) Teaching mathematics to adults.
[250]
Do we still need to teach fractions,
Proc. ICME 4.
Publications of P.J.
Hilton
13
[25l]
(with Joseph Roitberg) Restoration of structure, Cahiers Topologie Ge"om. Differentielle, 22 (1981), 201-207.
[252]
(with Jean Pedersen) On the distribution of the sum of a pair of integers.
[253]
(with Jean Pedersen) Casting out 9's revisited, Math Mag. 54 (1981), 195-201.
[254]
The emphasis on applied mathematics today and its implications for the mathematics curriculum, New Directions in Applied Mathematics, Springer (1982), 155-163.
[255]
Avoiding math avoidance, Mathematics Tomorrow, Springer Verlag (1981), 73-83.
[256]
(with Joseph Roitberg) Note on completions in homotopy theory and group theory.
[257]
(with Carl Bereiter, Joseph Rubinstein and Steve Willoughby) I. How deep is the water? II. Measuring bowser. III. Bargains galore, Thinking stories, Open Court Publishing Company, (1981), 91, 121, 121 (1981) .
[258]
(with Carl Bereiter, Joseph Rubinstein and Steve Willoughby) Real Math, Grade 5, Open Court Publishing Company (1981), 511.
[259]
(with Carl Bereiter, Joseph Rubinstein and Steve Willoughby) Real Math, Grade 6, Open Court Publishing Company (1981), 425.
[260]
Group structure and enriched structure in homotopy theory, Proc. Math. Sem. Singapore (1980), 31-38.
[261]
The language of categories and category theory, Math. Intelligencer 3 (1981), 79-82.
[262]
Group structure in homotopy theory and generalizations of a theorem of Blackburn, Atas.
[263]
Relative nilpotent groups, Proc. Carleton Conference on Algebra and Topology (1981).
[264]
Some trends in the teaching of algebra, Proc. Hongkong Conference on New Trends in Mathematics (1981).
[265]
The education of applied mathematics, SIAM News 14, 5, October (1981).
[266]
Homotopy, Encyclopedia Britannica.
[267]
(with Jean Pedersen) 501-502.
[268]
Nilpotent groups and abelianization, Questiones Mathematicae (1982) .
[269]
Groupes relatifs et espaces relatifs, P r o c , 6 i e m e Congres du Groupement des Mathematiciens d1Expression Latine (1982).
[270]
Review of 'Mathematics: The lose of certainty" by Morris Kline, Bull. London Math. Soc. (1982).
[271]
Reflections on a visit to South Africa, Focus, Math. Ass. Amer., November, (1981).
e^ > i\e ?, Mathematics Teacher 74 (1981),
Publications of P.J.
Hilton
14
[272]
ICMI, 1980-81, L'Enseignement Mathematique (1982).
[273]
Mathematics in 2001 - Implications for today's Undergraduate Teaching, Proc. Conf. Remedial and Developmental Mathematics in Colleges, New York, April, 1981.
[274]
(with Gail Young, ed) New directions in applied mathematics, Springer (1982).
[275]
Review 'A Brief course of higher mathematics, by V.A. Kudiyovtsev and B.P. Demidavich, Amer. Math. Monthly (1982).
ESSAY ON HILTON'S WORK IN TOPOLOGY Guido Mislin, ETH Zurich
INTRODUCTION Peter Hilton's work in topology covers a wide range of topics, some have a distinct geometric character, others are more algebraic. Often a basic idea is modulated through different keys, creating new variations of old concepts in a different category. A typical example is that of the topological notion of homotopy, which is transformed into the algebraic concept of projective and injective homotopy for modules. The resulting algebraic structures were studied jointly by Beno Eckmann and Peter Hilton. Passing back from algebra to topology led to the origin of the Eckmann-Hilton duality (cf. Urs Stammbach's article in this volume). Hilton's early work concerned mainly homotopy groups, Hopf invariants, Whitehead products and the homotopy groups of a wedge of spaces (Hilton-Milnor formula). In the first section of this essay we will concentrate on Hilton's paper "On the homotopy groups of the union of spheres" [H13], which is the perfect example of his clarity of exposition and his ability to use efficiently the interplay of algebra and geometry. The topics which follow are not meant to represent his entire work. They serve only as illustrations of different aspects of his mathematical accomplishments.
Mislin: Hilton's work in topology
16
The following table illustrates the main areas of Peter Hilton's work in topology.
Homotopy theory k
/
Hopf invariants
/ /
,
/
Eckmann-Hilton duality
Wedge an d product cancella tion
Group like structures
Exotic H-spaces
/ / L.S -category/ / and cocategcDry r
/ /
/ Kan extensions
Shape theory
/ General cohomology
Nilpotent spaces /
/
Localization and completion
Genus of spaces and bund les
Peter Hilton is not only a fine research mathematician, he is a great teacher and mentor. Of his more than thirty coauthors, many were young and inexperienced when they first worked with him? these beneficial professional collaborations have often lead to lifelong friendships. I happily count myself one of these fortunate people. Thank you Peter.
1.
THE HILTON-MILNOR FORMULA
A homotopy operation in
k variables is a natural
transformation TT
n
(X)
l
X
...
X 7T
n
(X)
k
+
7T
n
(X)
Mislin: Hilton's work in topology
17
Such operations correspond bijectively to the elements of n n n n 7Tn(S
v . . .vS
puted
IT (T)
) = TTn(T) ; T := S
for a
1-connected
T
v . . . vS
. Hilton com-
in [H13], expressing it
in terms of homotopy groups of spheres. His calculation makes use of the relationship between Whitehead product in TT^ (T) H A (OT)
TT^(T)
and H^(OT) , the
and the Pontryagin product in
. The following is a short outline of his beautiful
proof. By a result of Bott and Samelson [2] the Pontryagin algebra H^(ftT) = R
is a free associative algebra, freely generated
by elements n
S
n
l
k ,...S
rings
e ,...,e,
, which correspond to the spheres
via transgression: thus
R
is a coproduct of
2Z[e,] * ... * Z[e, ] , the generator
e. having degree
n.-l . A theorem of Samelson [16] implies that the composite map 7i n (T)
p[a,3] = (-D P (p(a)p(3) - (~D Pq p(3)p(a))
fulfills for
a e TT ,. (T) and $ eC TT+ (T) . This lead Hilton to dep+i 3 ' fine the quasi-commutator [a,b] for two elements a,b e R of gradation
p
and q
by setting
[a,b] = (-l) p (ab-(-l) pq ba) The map
p : '%(T) -> Hilr(^T)
of degree
-1 then fulfills
p[a,B] = [p(a),p(3)] . The additive structure of R =2Z[e,] * ... * 7L[ e, ] scribed as follows. By an inductive procedure, ducts in R weight
are defined and ordered. The basic products of
1 are the generators
ed by setting weight
e,,...,e, e R ; they are order-
e1 < e 2 < ... < e, . If the basic products of are defined and ordered, then the basic products
of weight
r
a
are basic products of weight
and b
can be de-
basic pro-
are defined to be the elements
[a,b] £ R
where
u and v respectively,
Mislin: Hilton's work in topology
u + v = r
a < b ,
/
and if
b = [c,d]
c £ a . The b a s i c products of weight
18
, then one must have r
are given an a r b i -
t r a r y order and they a r e considered g r e a t e r than basic p r o ducts of l e s s e r weight. If the monomial i < j
(£ m)
z ,...z
e R
are basic products,
z. . . . z i s said to have zero d i s o r d e r , if 1 m implies z^ £ z. . Hilton proved the following
crucial result,
g e n e r a l i z i n g ideas of M. Hall [ 6 ] , Ph. Hall
[7] and Magnus [ 1 3 ] . Theorem 1;
The monomials of zero d i s o r d e r in basic
products of elements of d i t i v e b a s i s for
R = 2[e,]
For each basic product W(TT) e TT^(T) sion class
* . . . * Z2[e, ] form an ad-
R .
w e R
is defined by replacing each i. : S
-> T
Whitehead product in
TT^(T) ; these elements
to
x
R
by the
W(TT) are called
O(W(TT)) = w . Thus, if n w
•+ T , the map .ft(w(ir)) : Q.S
ical generator
e. by the inclu-
and the bracket in
basic Whitehead products. Note that n w W(TT) : S
an element
->•ftTmaps the canon-
of the polynomial ring
H^(ftS w ) = 2Z[x]
w £ H^(ftT) . It follows then in view of Theorem 1
that
the product map ITftSW •+ QT w induces a homology equivalence; it is therefore a homotopy equivalence, because the spaces in question are H-spaces. Hilton's theorem can now be stated as follows.
Theorem 2:
-n^S v...vS
) =
0 ,n(snw)
W(TT) n
where
W(TT) e ^^(S
l
n
v...vS
k
)
runs over all basic Whitehead n products and where the direct summand IT (S w ) is embedded
Mislin: Hilton's work in topology n
by composition with
W(TT):S
n
w
•* S
l
n
V . . . V S
k
The theorem may be used to give a short proof for the "Jacobi Identity" for Whitehead products. Setting L (X) = 7T _ (X)
and using the Whitehead product as multipli-
cation, one obtaines a graded ring ted space
X . This ring
L^(X) for any
1-connec-
L^(X) is a quasi-Lie ring in the
sense of Hilton. He studied the structure of quasi-Lie rings in general and determined the additive structure of free quasi-Lie rings in [H20]. It turns out that the only relations which hold in
L^(X) for all
X , are the relations which
hold in a free quasi-Lie ring. Theorem 2 was generalized by Milnor in the following way (cf. Adam's student guide [ 1] ) .
Hilton-Milnor Formula Let
X ,...X,
QZ (X v X o v ... v X, ) 1
Z
be connected
CW-complexes. Then
has the same homotopy type as a weak
K
infinite product
ff fiZX. , where each DX. , j > k , is an D 3* 1 (n-,) (n v ) iterated reduced join X^ x \ ... A X, . The number
of factors of a given form is equal to I n
z
P(d) (n/d)!
=
d/6
6 = gcd(n,,...,n, )
+
1
and
y
k
denotes the Mobius function.
The following application is contained in Hilton's papers [H79] and [H82]. Let L P types of
denote the set of homotopy
1-connected finite polyhedra, which are homotopy
equivalent to suspensions. Consider the wedge operation, and denote by Grothendieck group; write by
X
in
1
[X]
E3P
as a monoid, using
G(£P )
the associated
for the element represented
Mislin: Hilton's work in topology Theorem 3;
Let
X
and
20
Y
be
1-connected
poly-
hedra, which are of the homotopy type of suspension spaces. If
[X] - [Y] e GCCIP1)
is a torsion tor element, then
X
and
Y
have isomorphic homotopy groups.
For example, if
X,Y
and
A
finite suspension spaces such that follows that
7ri(X) = TT 1 (Y)
are
1-connected
X v A - Y v A , then it
for all
i , since
[X] = [Y]
in this case; a generalization of this result is due to Kozma [ 1 2 ] , a student of Hilton. Note that it is still an open question whether
G(ZP )
possesses any non-trivial torsion
elements.
2.
THE HILTON-HOPF
INVARIANTS
The classical Hopf homomorphism (cf. [8]) morphism
H : TT
I ( S ) "* 7L
was generalized first by G,W. Whitehead to a homo H :
7T
n (S
r
) +
7T
Hilton to groups with
n(
s2r
"" 1 ) '
n £ 4r-4
n
<
3r
"3
•
and
later, by
(cf. [H3]). Hilton's defini
tion is as follows. One considers the map H* : TTn(Sr) ^TT n (S r v S r ) ^ T T n + 1 ( S r x S r , S r v S r ) - * V where point,
$
is induced by pinching the equator of Q
Sr
( S
to a
is the projection onto a direct summand and
induced by shrinking
S
v S
}
x
is
to a point. Note that the Freu-
E : TT (S ) •> TT . (S r ) is an isomorn n+i n £ 4r - 4 . The map H is then given by
denthal suspension phism for
H = E * 1 o H* : TT (S r )+ TT (S 217 " 1 ) , n £ 4r - 4 . n n In [H13] Hilton proposed the following natural generalization of the homomorphism
H . Recall that (Theorem 2)
Mislin: Hilton1 «* work in topology
n
21
0 1Tn(Sr) © ^ ( S ^ " 1 ) © (TTn(S3r""2) © T^ (S^""2) ) © ...
n
IT (S r v S r )
where the right hand summands are embedded in
by composition with basic Whitehead products. Let i = 1,2,..., denote the projection onto the r
and set H. -, =H\ . o $ , $ : TT (S ) •> TT (S i~i l-l n n This defines the Hilton-Hopf homomorphisms
H
1'H2
:
V
Hilton proves, that H , and
E oH
3
^ * V(s3r"2)
H
r
**'_•»
(i+2)-nd summand v Sr)
as above.
••••
agrees with the previously defined
equals
H* . The Hilton-Hopf homomorphisms
may be used to express the left distributive law for composition of homotopy classes as follows. Let by definition, one obtains in
TT (S V S )
y e TT (Sr) . Then, the following
equation
oH 1 (y) + [i2,[i1,i2]] oH 2 (y) where
i,
and
inclusions
S
r
i
denote the homotopy classes of the two r r -> S v S . Thus, if a, 3 e TT (X) and
Y e ^n(Sr) , (a+3) o y = a o y + $ o y+ [af 3] o H Q ( Y ) +
[a,[a,3]] oH1(y)
+ [3, [a , 3] ] o H 2 (y)
+
...
The Hilton-Hopf invariants measure therefore the deviation from additivity for the homotopy operation y* :
TT
(X) •>
TT
(X) , a »> a o y = y*(a)
Mislin: Hilton's work in topology
For
22
k £ 7L one obtains the simple formula k
lr 4-1
Y*(ka) = ky*(a) + (*) [a,a] o H Q (y) + 2 ( 3 ) [a, [a,a]] o H ^ y ) and the last term, involving
[a,[a,a]] , vanishes if
r
is
odd.
3.
ECKMANN-HILTON DUALITY
In the Comptes Rendus notes [H23,H24,H25] Eckmann and Hilton presented for the first time a frame work for an internal duality in the pointed homotopy category. Their starting point was the homotopy set sidered as a functor in
[X,Y] , which can be con-
X , or "dually" as a functor in
Y .
They introduced in a self-dual way homotopy groups 7Tn(X;Y) = [Z n X,Y] = [X,ftnY]
generalizing simultaneously co-
homology groups and homotopy groups; by relativizing with respect to either variable they obtained dual long exact sequences, and the term "cofibration" appeared, dual to the notion of a fibration. In view of this duality, a pair of 'spaces should be considered as a map, and a triad as a diagram X •> Y -+ Z . This point of view leads to dual triple sequences in homotopy and cohomology, which were used by Eckmann and Hilton [H30] to define the homotopy and homology decomposition of a map. In later papers [H51,H56,H57] they studied internal duality in arbitrary categories. In particular, they analysed the concepts of groups and cogroups in a general setting. The reader who wishes to find out more about the history and meaning of duality is referred to Hilton's recently published review article [H242]. We will try to give an idea of EckmannHilton duality by looking from a dualists point of view at a specific example.
M i s l i n : H i l t o n ' s work i n t o p o l o g y
23
To fix ideas, we place ourself topy category of connected
in the pointed homo-
CW-complexes. An
a space for which the folding map
H-space
X
is
V : X v X -* X extends to
t X v
Dually, a map
coH-space
A : Y -> Y x y Y v Y
Y
is a space for which the diagonal
may be compressed into Y v Y :
<~-J
Y x By a result of James [10], the canonical map is a
X
is an H-space if and only if
X -> ftEX has a left inverse. Dually,
coH-space if and only if the canonical map
y
Y.QY •> Y
has a right inverse; this has been proved by Ganea [4]. James proved his theorem using the "James model" for ftEX ; there is no duality visible relating his proof to Ganea's. It is well known that every y : X x x •+ X
H-structure
admits an inversion on each side: we say
(X,y)
is a loop-object. This may equivalently be expressed by the fact that the two shift maps X x x
lxA ->
X x x x x
y x l + X
are homotopy equivalences for every But not every
coH-structure
x
x
,
a n d
H-structure
v : Y -> Y v Y
lyv -*
Y v Y v Y
Vvl •*
on X .
admits inverses,
that is, the corresponding maps Y v Y
y
Y v Y , and
Mislin: Hilton's work in topology
Y v Y
vvl ->
Y v Y v Y
24
lvV -> Y v Y
need not be homotopy equivalences: there are coH-structures such that (Y,v) is not a coloop-object (examples are due to Barratt). However, if Y is 1-connected, one can dualize the proof for H-spaces and show that for every coH-structure v on Y , (Y,v) is a coloop-object (cf. [H218]). It is an elementary fact that every finite H-space X is of the form S * Z , where S is a product of circles and H (Z) = 0 . Dually, one might expect the following to hold (cf. [5]). Ganea Conjecture: Let coH-space. Then Y - T v W with
Y T
be a finite (connected) a wedge of circles and
TTX(W) = 0 .
The Ganea conjecture has been proved for coloopobjects [H218]. Note that no coH-spaces Y are known which do not admit a coloop structure Y -> Y v Y .
4.
THE HILTON-ROITBERG MANIFOLD AND LOCALIZATION
Hopffs original paper [9] marked the beginning of the study of topological groups from a homotopy point of view. When, more than 25 years later, Hilton and Roitberg found the first example of a finite Hopf-space different from the obvious ones (Lie groups, S , P-OR) and their products), the study of H-spaces became a very active field of research. The new H-space, which was denoted by E_ , was shown by Stasheff to be of the homotopy type of a topological group [18]. One can describe E_ as follows. Consider the princi7w 3 7 3 7 pal S -bundle Sp(l) *+ Sp(2) •* S and let S •> E -> S
Mislin: Hilton's work in topology
denote the bundle induced by a map Then, as we will sketch below, and
Sp(2) x s
manifold group
E-
25
S
•> S
of degree
E_ ? Sp(2)
but
n .
x S3
E_
are diffeomorphic. This implies that the is an
H-space, being a retract of the Lie
Sp(2) x s The basic construction behind this type of example
was first considered by Hilton [H82] in a dual setting, where he demonstrated the failure of wedge-cancellation. His dual a, 3 £ TT _, (Sn)
examples are as follows. Let Cg
be the mapping cones of
a
and
and let
Blakers-Massey Theorem, it follows that
C
- CD ex
if
±3 = (±1) o a . Suppose now that
and that
3 = &cx , with
£
prime to
C
,
3 . Using a form of the
a
if and only
p
has finite order
k
k . By forming the push-
out diagram a
S where
i
and
xft
C
o CXp
- C
v S
«B
are the embeddings of
cones, one infers fore
+C
i o 3 = i o £cx = £(i m
. Similarly,
C
CX
o CXp
S
in the mapping
oa)=0,
- CD v S
m
and there-
, since
p
v S m - Co v S m . The resulting ex p non-cancellation example of lowest dimension is as follows. Let co e IT _ (S ) be of order 12. Then a = £3
for some
I . Thus
C
b
^
S
7
V e (i)
^
7
? S
V e 7co
^
, b u t
(S
7
\J e ) v S co
7
-
^
( s V e ) 7co
7
7
v S
By passing to the dual situation, one obtains examples for non-cancellation of factors in a product. But these examples will not be finite dimensional, since the EckmannHilton dual of a sphere
Sn
is an Eilenberg-MacLane space
K(Z,n), which is of infinite dimension for
n > 1 . It is
Mislin: Hilton's work in topology
26
therefore necessary to modify the dual situation. Hilton and Roitberg proceeded as follows. They considered principal bundles
G •> E
-> S
n
, classified by
nected Lie group. Supposing that with
I
E
and 3 = la
is defined by the pull-back diagram
E
Q
"*•
a3
The maps
E
D
3 IP 3
\
p
and p g
are the natural projections, and a*
denotes the adjoint of
a e TT _, (G) . Of course,
3* = &a* . One would like to show that
which would imply structure in to
k
x G - E n x G - E o , 3 a3
a
and
has order
a con-
prime to k , one wishes to infer E
where
a
a e TT . (G) , G
G-
E
R
[E ,BG]
- E
a* o p
= 0,
la* o p ^ = 0 ,
x G . The lack of a natural group
makes it difficult to compare
la* o p
a* o p
. Hiljbon and Roitberg found the following theorem 3 to deal with the case G = S (more general results may be found in the later papers [H15O,H161,H186]). a
Theorem 4: Let a e TT wol
a = 0 , where
•* Sp(2) -»• S
. Then
If we take since
Tr« (S ) 3
E
x s
- E_
E
J- E_
OJ e TT - (S )
la* o p
n = 7
x s
noj
classifies
and a = a) , then
; by definition
7o)* o p E
= 0
3 ( w o l a) = 0 and therefore
= Sp(2) . To see that
, one makes use of the homology decompositions of
these spaces. By a result of [111. E
and assume that
= 0 .
has order 3 . Thus 3
(S )
-S
served e a r l i e r ,
KJ e Kj e no)
I.M. James and J.H.C. Whitehead
. Thus
E a)
S 3 U e 7 f S3 V e 7 . a) 7o)
? E_ 7o)
since, as we ob-
Mislin: Hilton's work i n topology
27
The non-cancellation aspect of this example as well as the construction of new H-spaces become much more transparent, if one uses localization techniques. The localized fibrations S
P "*
(E
^ P " SP '
a n d
7 Sp3 -> (E_ ) -> S 7urp p
turn out to be fibre homotopy equivalent for each prime p ; the original fibrations belong thus to the same genus in the terminology of [H152]. In the more general setting of quasiprincipal G-bundles E. •+ B ,
i = 1,2
of the same genus, one can show that E, x F - E x F (cf. [H161]? "quasi-principal" means that the bundle projection p. : E. •> B composed with the classifying map B •* BG of the associated principal G-bundle is 0-homotopic). Recall that the genus set G(X) of a nilpotent space X of finite type consists of all nilpotent homotopy types Y of finite type with p-localizations X - Y for all primes p . For instance G(Sp(2)) = {Sp(2),E_ } . The /(JL)
genus sets of Lie groups provide a natural source for new Hspaces. Indeed one can show that if X is a finite H-complex, then every Y e G(X) is a finite H-complex (in [H186] it is proved that such a Y is a retract of X x X ; in the nonsimply connected situation one needs to know that the Wall finiteness obstruction vanishes for Y , cf. [15]). A careful analysis of the construction of the members of a genus set reveals that if X is a loop space, then every Y e G(X) is a loop space [H186]. Therefore, all members of the genus G(L) of a Lie group L have the homotopy type of topological
Mislin: Hilton's work in topology
28
groups. The genus sets of the simply connected rank two H-spaces are completely known. An interesting example is provided by the exceptional Lie group set of Y-3
G
and
(cf. [H152]). The genus = G
, Y
,
Y. , whose cartesian powers are related by 2 Y - YY 2 f ?Y Y 2 -Y Y 2 Y l 2 3 4 '
The
G
consists of precisely four members, Y
Y
4 Y4 Y Y 2 3
H-spaces of a fixed genus give always rise to non-can-
cellation examples of the type encountered in the original example of Hilton and Roitberg. One can show that if X
, X~ £ G(X) , X
a finite
product of spheres X and
x s - X X2
H-complex, then there exists a
S , depending on
X
only, such that
x S ; conversely, if the finite
stay in the relationship
X
H-complexes
X
x s - X 2 x s , with
produc a product of spheres, then one necessarily has
G
(xx)
S
=
cf. [14; After the first examples of new
H-spaces were known
and understood, it was natural to study the homotopy classification of
H-spaces with few cells. Peter Hilton was also
involved in this project [H97,H112,H152]. His joint paper with Roitberg [H112] contains the following complete list of homologically torsion-free rank two guity:
S
1
x S
1
, S
1
x S
3
* S
1
x S
7
H-spaces, modulo one ambi, S 3 x S 3 , SU(3) ,
S 3 x s 7 , Sp(2) , (E 2 J
, E3a) , E4a) , E ^
The one doubt concerns
E_ 2(A)
both or neither
and
Er
, (E^) , S7 x
S
7
.
, which were known to be
DO)
H-spaces; they are not, as was proved by
Zabrodsky [19] and, independently, by Sigrist and Suter using methods of
K-theory [17]. At this point, it would seem natural to add a sec-
tion on Peter Hilton's work in nilpotent spaces, which is intimately related to his current research in nilpotent groups. For instance, Hilton and Roitberg proved recently that nilpotent complexes of finite type are "Hopfian objects" in the
Mislin: Hilton's work in topology
29
homotopy category, generalizing the well known fact that finitely generated nilpotent groups are Hopfian. Influenced by a paper of J.M. Cohen [3], Hilton is studying jointly with Castellet and Roitberg "pseudo-identities", with applications to self maps of nilpotent spaces and homologically nilpotent fibrations. This work is still in progress and we will certainly hear more about it in the future.
REFERENCES For the references labeled H see the complete list of Hilton's publications in this volume. [1]
J.F. Adams, Algebraic Topology - A Student's Guide, London Math. Soc. Lecture Notes Series 4, Cambridge University Press 1972.
[2]
R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 320-337.
[3]
J.M. Cohen, A spectral sequence automorphism theorem; applications to fibre spaces and stable homotopy, Topology 7 (1968), 173-177.
[4]
T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970), 185-197.
[5]
T. Ganea, Some problems on numerical homotopy invariants, Lecture Notes in Math. 249, Springer 1971, 13-22.
[6]
M. Hall, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575-581.
[7]
P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1934), 29-35.
[8]
H. Hopf, Ueber Abbildungen von Spharen auf Spharen niedriger Dimension, Fund. Math. 25 (1935), 427-440.
Mislin: Hilton's work in topology
[9]
30
H. Hopf, Ueber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerung, Ann. of Math. (2) 42 (1941), 22-52.
[10]
I.M. James, Reduced product spaces, Ann. of Math. 62 (1955), 170-197.
[11]
I.M. James and J.H.C. Whitehead, The homotopy theory of sphere bundles over spheres: I, Proc. London Math. Soc. 4 (1954), 196-218.
[12]
I. Kozma, Some relations between semigroups of polyhedra, Proc. Amer. Math. Soc. 39 (1973), 388-394.
[13]
W. Magnus, Ueber Beziehungen zwischen hoheren Kommutatoren, Journal fur Math. (Crelle) 177 (1937), 105-115.
[14]
G. Mislin, Cancellation properties of H-spaces, Comment. Math. Helv. 49 (2) 1974, 195-200.
[15]
G. Mislin, Finitely dominated nilpotent spaces, Ann. of Math. 103 (1976), 547-556.
[16]
H. Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. of Math. LXXV (1953), 744-752.
[17]
F. Sigrist and U. Suter, Eine Anwendung der K-Theorie in der Theorie der H-Raume, Comment. Math. Helv. 47 (1) 1972, 36-52.
[18]
J. Stasheff, Manifolds of the homotopy type of (nonLie) groups, Bull. Amer. Math. Soc. 75 (1969), 998-1000.
[19]
A. Zabrodsky, The classification of simply connected H-spaces with three cells, II: Math. Scand. 30 (1972), 211-222.
THE WORK OF PETER HILTON IN ALGEBRA Urs Stammbach, ETH Zurich
Peter Hilton has published more than 250 articles and books? about 70 or 80 of them may be said to be of an algebraic nature. To give a description of the content of these papers in a few minutes is an impossible task. I shall therefore concentrate in this talk on a few aspects of Hilton's papers on algebra and try to describe the main ideas and the main results. I will not say anything about the numerous books and lecture notes Peter Hilton has written on various algebraic topics, although they undoubtedly constitute an equally important and influential part of his work. Many of the papers I shall be concerned with have one or sometimes two coauthors. Indeed, those who have had the chance to work with Peter Hilton know that he has an exceptional ability to stimulate and generate joint work. For those who know Peter Hilton it is no surprise that most of his papers on algebra have a close relationship to algebraic topology? indeed either the motivation for the paper or the applications or even both lie in topology. Thus this part of his work can only be described in relation to topology and the assignment of a paper to algebra or topology is often arbitrary. The first topic I want to talk about in some detail, the homotopy theory of modules, illustrates this point very clearly. The basic idea leading up to the theory was conceived in Zurich in 1955. Hilton, in his retrospective essay on duality in homotopy theory [H242], relates the story. In Warsaw Hilton had met Borsuk and had learned from him his definition of de-
Stammbach: The work of Peter Hilton in algebra
pendence of (continuous) maps between g : X ->• Z complex
is said to depend on
X
f : X •* Y
containing the map
X
g
f
if for every
g
h : Y -> Z
depends on
such that
g
CW-
can be extended to
can be extended to a map
Borsuk had proved that there exists
CW-complexes. The map
f :X + Y
such that
32
f
g : X ->• Z .
if and only if
is homotopic to
h of .
In the same year Hilton went to Zurich where on a suggestion of Beno Eckmann the two studied the same question for A-modules and in particular gave a sensible definition of homotopy of module maps. Since maps between modules may be subtracted, it is enough to define nullhomotopy. Borsuk's result suggested one define a homomorphism
$ : A -* B
nullhomotopic if it extends to every supermodule
A
to be of
A .
It is easy to see that this is equivalent to the requirement that
< ( > factors through an injective supermodule
A
of
A .
The homotopy of module maps defined in this way was called injective or
i-homotopy. The duality in module theory sug-
gested then also a definition of projective homotopy: the map
is called
projective module
P
p-nullhomotopic if it factors through a having
B
as quotient. It is inter-
estinq to note that Eckmann and Hilton have never published a joint paper on homotopy theory of modules? there are only two separate publications, namely transcripts of two talks, one given by Beno Eckmann in Louvain [3] and the other given by Peter Hilton in Mexico [H31]. In addition there is a short treatment of the theory in the book by Hilton on homotopy theory and duality [H77*]. These facts perhaps explain why the homotopy theory of modules has not become so widely known and employed in homological algebra as one might have expected. Let me briefly try to describe the main ideas.
If topic maps
A,B
are two A-modules, then the
< | > : A •* B
form a subgroup
Horn (A,B)
i-nullhomoof
Horn(A,B) . As in topology Eckmann and Hilton define the ihomotopy group
TT(A,B)
by
Stammbach: The work of Peter Hilton in algebra
33
TT(A,B) = Hom(A,B)/Hom o (A,B) .
If
A
is an
injective
module
containing
A
(an
"injective container" as Peter Hilton calls it in [H77*]) and A >-> A ->> A/A
the corresponding short exact sequence, then
again as in topology A/A
a suspension
motopy group
A
EA
is called a cone
of
TT(A/A,B)
CA
of
A and
A . It is easy to see that the ho-
is independent of the chosen injective
A . This leads then to the
n-th i-homotopy group by setting
ifn(A,B) = 7F(InA,B) where
EnA
is the
n-fold suspension of
A . For
n > 2 we
have Trn(A,B) = Hn-.1(Hom(I,B)) where
I
is an injective resolution of
fine relative
A . One can then de-
i-homotopy groups and one obtains the usual
exact sequences. In particular it is easy to see that any short exact sequence
A 1 >-> A - » A"
of modules gives rise
to an associated long exact sequence '/B) + TTn(A",B) + TTn(A,B) + 7Tn(A\B) -• ... (A'.B)
+ 7To(A"fB)
-> T T O ( A , B )
+ Uo(A',B)
•> E x t 1 ( A I I f B )
+ ...
. . . -> Eactn"1(AlfB) -> Extn(A",B) •> Extn(A,B) ->Ext n (A l / B) ^ . . . In
[4 ] Eckmann and Kleisli have shown that in the case where
A
is a Frobenius algebra this sequence is essentially the
complete homology sequence. In 1960 a paper by Heller appeared [7] where analogous results for the homotopy theory in an arbitrary abelian category are obtained. (Since Heller works with
p-homotopy he
has to suppose that the category has enough projectives.) The notation ftA for the kernel of a projective presentation
Stammbach: The work of Peter Hilton in algebra
P ->> A
of
A
34
was the same as that used by Eckmann and Hil-
ton. In 1961 Heller published a further note [8] where he proved that in a category of modules over an Artin algebra, the fact that
A
is indecomposable implies that
Q.A is in-
decomposable provided the projective presentation is minimal. This remark has become extremly important in representation theory, and
Q(-)
is nowadays called the Heller operator.
In a certain sense the work on homotopy theory of modules was continued by Peter Hilton in a joint paper [H49] with Rees in 1961. Here the authors work in an abelian category
Q
with enough projectives and specialize to an appro-
priate category of modules only if necessary. This more general viewpoint had been made possible by developments between 1956 and I960, in part by joint work of Hilton and Ledermann [H22], [H32], [H34]. They defined a categorical structure, called a homological ringoid, in which the basic notions of homological algebra could be defined. But let me return to the paper by Hilton and Rees. Their aim was to obtain an elementary homological proof of a result of Swan: A group
G
periodic cohomology of period
has a
q
if and only if
G-projective resolution of period functors
Ext_(ZS,-)
and
i > 1 .
has
q . Of course, a group
is said to have periodic cohomology of period valent for all
2Z
Ext^ ^(2Z,-)
q
G
if the*
are naturally equi-
This led the authors to consider
quite generally the natural transformations between Ext p (B,-) , B e Q from for
Q
and
F(-)
where
F
is an additive functor
to the category of abelian groups. They showed that
p > o Nat(Ext P (B,-) , F(-)) = S F(-) ,
where
S
P
F
(~)
denotes the
p-th left satellite of
F . This
is a far reaching generalization of what is known as Yoneda's lemma (which appears as Lemma 1.1 in this paper). For
F(-) =
A <S> - , this gives a characterisation of the Tor-functor
Stammbach: The work of Peter Hilton in algebra Tor (A,B) = N a t ( E x t P ( B , - ) , A 0 - ) and for
F(-) = Ext 1 (A,-)
TT(A,B)
i s the
,
i t yields
Nat(Ext 1 (B / -), Ext 1 (A,-))
where
35
= TT(A,B)
p-homotopy group of Eckmann-Hilton.
In modern language one would describe this
last
r e s u l t as follows. The assignment A *^> Ext 1 (A,-)
defines a (contravariant) functor from the category Q to the category of additive functors from C to abelian groups. The objects Ext (A,-) may be identified with the p-homotopy classes of objects in Q (in the sense of Eckmann-Hilton) or the classes of quasi-isomorphic objects (in the sense of Hilton-Rees). The morphism group between two such objects, i.e. Nat(Ext1 (B,-) , Ext1(A/-)) is TT(A,B) , the p-homotopy group of Eckmann-Hilton. I wanted to include this modern description of part of this paper, since in 1975 AuslanderReiten [1] used this functor and the above result (as well as many other things) in their proof of the existence of almost split sequences. These sequences have proved in the mean time to be an indispensable tool in modern representation theory of Artin algebras. Let me go back to homotopy theory. I have tried to explain how an attempt to copy certain ideas in topology into module theory led to the interesting notion of i-homotopy of modules. I have already also said that in module theory there is an obvious dual notion, that of p-homotopy. One of the key ideas of Eckmann-Hilton was to translate the dual notions in module theory back into topology. This led to the celebrated Eckmann-Hilton duality. But this really is topology, and no doubt a substantial part of Mislin's essay on Hilton's work in topology [10] will be devoted to a discussion of that
Stammbach: The work of Peter Hilton in algebra
36
duality. What is important here, is that this idea also gave rise to a development in category theory. I mean of course the series of papers, again jointly with Beno Eckmann on group-like structures [H51], [H56], [H57]. Starting from the wellknown fact that the set of homotopy classes of maps TT(X,Y) form a group if Y is a "group up to homotopy11, the authors developed the notion of a group-like structure in a general category Q . A group-like structure on the object C in C is a morphism m : C x C -> C which makes the set £(X,C) into a group,naturally in X . In the three papers the two authors adopted an abstract categorical approach. The return they gained from this was enormous. First of all it made available to them the general duality principle of category theory, thus putting for example the notion of a cogroup in a category on the same level as the notion of a group. Secondly, it directed attention, in general and in applications primarily to those functors which respect certain constructions and which therefore carry the group-like structures from one category to another. This proved very fruitful from a heuristic point of view. As an (easy) example of a result that may be found in these papers I mention the following. If X is a cogroup in Q and Y is a group in Q , then Q(X,Y) inherits two natural group structures, one coming from X and the other coming from Y . It is shown in these papers that quite generally the two group structures agree, and that they are commutative. In particular, it follows that different cogroup structures on X and different group structures on Y give rise to one and the same abelian group g(X,Y) . This of course generalizes the fact that the fundamental group of a topological group G is abelian and that its group operation may be defined using the group operation of G . In the course of these three papers the authors introduced and discussed many basic notions of category theory,
Stammbach: The work of Peter Hilton in algebra
37
like direct and inverse product, equalizer and coequalizer, intersection and union, adjoint functor etc. In reading these papers again I was reminded of my student days, when I learned a lot of category theory from them. We referred to these papers as the "trilogy by the pope and the copope". It is perhaps one of the nicest comments which can be made about a mathematical paper, that twenty years after its publication its content is common knowledge of working mathematicians. Such a comment certainly applies to these three papers. The next algebraic topic Peter Hilton addressed was the theory of spectral sequences. Again jointly with Beno Eckmann he published a series of papers on the subject. Spectral sequences had been around for several years and had proved to be a very important tool in both topology and algebra. But a readable account of the theory that covered the many formally distinct cases was still missing. The basic idea was distilled from their previous work on homotopy and duality. Basic to that theory is the notion of a composition functor T , i.e. a graded functor defined on a category of maps to the category of abelian groups and such that a factorization of a map f , f = g o h gives rise to an exact sequence T q (g) -* T q (f) •> T q (h) - T q _ 1 (g)
In [H68] the authors generalized this and considered infinite factorizations, f = ••• o J p+ i o 3 p o j p _ 1 o .... Obvious examples are the skeleton decomposition of a simplicial complex, or the Postnikov decomposition of a 1-connected space. The authors showed that such a factorization, together with a composition functor gives rise to a spectral sequence. It is obtained via the associated exact
Stammbach: The work of Peter Hilton in algebra
38
couple and the Rees system. The treatment leads to results even in case the spectral sequence does not converge in the usual sense. In that case the E^-term of the spectral sequence is not isomorphic to the graded group associated with T(f) , but the theory allows to identify the deviation from an isomorphism. A special feature of the authors approach to spectral sequences is that they work in an ungraded category so long as possible and only introduce the grading where it is really needed, for example for the discussion of convergence questions. This not only facilitates the presentation enormously, but also makes the whole theory more transparent. A student can now learn the subject without getting lost in a sea of indices. In order to be able to subsume all the spectral sequences under one and the same theory they worked in an arbitrary abelian category [H70]. This had the further advantage that full use of the duality principle could be made. However in order to be able to proceed in this generality, the authors had to solve certain new problems of a more categorical nature? some of these have given rise to separate publications. In [H74] for example the notions of filtration, associated graded object and completion are discussed in great detail. Here direct and inverse limits are introduced in the framework of Kan's theory of adjoint functors. In [H86] they examine the curious fact that the E^-term of a spectral sequence is obtained as a limit followed by a colimit and also as a colimit followed by a limit. In general this procedure does not yield the same object? it does however in the special case that appears in the theory of spectral sequences. This question led to problems in category theory involving pushout and pullback squares. For these questions Peter Hilton was well prepared, for in 1965 at the conference on categorical algebra in La Jolla he had given a talk on "Correspondences and Exact Squares", a topic which is related to pullback (cartesian) and pushout (cocartesian) squares [H73]. In trying to describe
Stammbach: The work of Peter Hilton in algebra
39
the higher differentials of a spectral sequence Peter Hilton was led to consider correspondences (relations) between objects of an abelian category £ . These define a new category A . By introducing the device of an exact square in £ Hilton was able to give a lucid description of the algebra of correspondences and of the way in which A is obtained from £ . He showed that there is a strong analogy with the classical procedure for passing from integers to fractions. Indeed, composition of morphisms in % for example, corresponds to multiplication of fractions, etc. I would like to add a further remark on this paper. It is the only research paper I know of which contains a reference to work done in elementary school. The reference is to a project of how to teach fractions to children. This then perhaps is a good place to say that it is impossible to fully appreciate the mathematician Peter Hilton without taking into account his concern with all aspects of education in mathematics, be it in elementary school or in graduate school. His publications and his lectures are vivid evidence of his conviction that teaching mathematics is an integral part of doing mathematics. Again I break the chronological order to mention a further paper on spectral sequences. In [H165] of which I am a coauthor the naturality of the Lyndon-Hochschild-Serre spectral sequence for the homology of a group extension is exploited to yield results on the torsion of the differentials of the spectral sequence. Let A >-> G ->> Q be a group extension with abelian kernel and let £ : A •+ A be multiplication by the natural number I , then the diagram £ :
A >-> G ->> Q
A >-> G ->> Q
Stammbach: The work of Peter Hilton in algebra
40
induces a map between the two associated spectral sequences. In particular, if £*U) = C one obtains an automorphism of the spectral sequence. This can be used to deduce properties of its differentials. As an example I mention the case, where the extension splits. Then clearly £ = 0 = &*(£) . Hence the following square is commutative
Hi(Q,H1(A,S)) + H±_2(Q,H2(A,S)) d
H±(Q/H1(A,ZZ))
2
Choosing I = 2 and noting that 2* on the left is multiplication by 2 and 2^ on right is multiplication by 4 one obtains 2d2 = 0 . This result had been proved earlier by Evens [5] and Charlap-Vasquez [2] by rather complicated arguments. This technique and extensions of it produced various results on the torsion of the differentials of the Lyndon Hochschild-Serre spectral sequence and eventually also estimates on the size of certain homology groups. In another series of papers Peter Hilton was concerned with central group extensions and homology. If N >-> G ->> Q is a central extension, then Ganea had shown using topological methods that the familiar five term sequence in integral homology can be extended by one term to the left: (*)
N © G a b - H2G - H2Q - N -> G a b - Q a b - 0 .
In the first paper of the series [H140], jointly with Eckmann, the spectral sequence of the fibre sequence K(G,1) -* K(Q,1) -> K(N,2) was used to obtain a longer exact sequence in homology start-
Stammbach: The work of Peter Hilton in algebra
41
ing with H^Q and containing Ganea's result as a corollary. A second line of attack is described in a paper by Eckmann, Hilton and myself [H136] where the map x ' N ® G , -> H2G is identified with a commutator map in a free presentation of G . For extensions of a special kind (weak stem extensions) the sequence (*) was extended two further terms to the left (**)
H3G - H3Q - N ® G a b - H2G - H 2 Q - N - Gafa - Q a b - 0 .
This opened the way to an efficient and elegant approach to Schur's theory of covering groups, including Kervaire's results. In a second paper [H139] of the three authors a variant of the sequence (**) for arbitrary central extensions was obtained and in [H155] the sequence (*) was used to yield information about H-Q , if G is a direct product, so that B^G may be assumed to be known. In [H162] Hilton and I worked on the problem of obtaining a sequence analogous to (*) without the hypothesis that N be central. Nomura [11] had already obtained such a sequence by topological methods. We proceeded algebraically by exploiting the naturality of the Lyndon-Hochschild-Serre spectral sequence and were able to generalize Nomura's result to arbitrary Q-coefficients. By a further analysis of the spectral sequence we also obtained an exact sequence for central extensions starting with H,Q , different from the one described in [H140]. I would finally like to remark that these sequences for central extensions which I have mentioned above as well as their interrelationship have been carefully discussed in a paper by Gut [6]. A long series of articles of Hilton with various coauthors is devoted to the study of localizations of nilpotent groups and spaces. In the early seventies Bousfield-Kan and Sullivan defined a localization for topological spaces. The proper setting for this theory is the category of nilpotent spaces? a space X is called nilpotent if its fundamen-
Stammbach; The work of Peter Hilton in algebra
42
tal group TT-T(X) is nilpotent and acts nilpotently on the higher homotopy groups IT . (X) , i > 2 . If X = K(G,1) with G a nilpotent group, the localization of X at a family P of primes is an aspherical space X p . Hence X p = K(Gp,l) for some group G p , the (algebraic) P-localization of G . If G is abelian, G p is nothing else but the ordinary localization of G , known from commutative algebra. Various algebraic questions suggested themselves; I will only be able to mention a few. Firstly a connection between localization and homology surfaced, in that a nilpotent group G is Plocal, i.e. isomorphic to its own P-localization if and only if its integral homology in positive dimensions is P-local. Starting from this Peter Hilton in [H167] described an elegant homological proof of the existence of the P-localization G p of a nilpotent group G by an induction on the nilpotency class. In his approach he used as part of the inductive process the key fact that the natural map £ : G •> G p is a Pequivalence, meaning that ker I is a P'-group and that there exists to every x e G p a P'-number n such that x n e &(G) • In further papers, some of them joint, Peter Hilton addressed the following questions: Which group theoretic constructions are compatible with the localization map I ? What are the classes of groups in which a localization with good properties can be defined? What is the relation of localization to other group theoretic notions like the notion of the isolator of a subgroup? Let me finally briefly describe the content of the two papers [H187], [H188], which are the product of joint work with Guido Mislin. Here some remarkable squares of nilpotent groups are discussed, namely squares of homomorphisms in the category N of nilpotent groups
H + p L
Stammbach: The work of Peter Hilton in algebra in which
$ , a
valences where P v Q = TT
are P
43
P-equivalences and ty , p
and
are
Q-equi-
Q are two families of primes with
the family of a l l primes. Such squares arise natu-
rally in localization theory, for whenever one has a nilpotent group
G the localizing maps form such a square G \
+ Gp 4-
G
Q " GP
In [H187] Hilton and Mislin show that every such square is simultaneously a pullback and a pushout square in g . This is rather surprising since in Jj pushout squares do not in general exist. This result was applied in [H188] to the study of the genus of a finitely generated nilpotent group with finite commutator subgroup. If N genus
G(N) is defined as the set of isomorphism classes of
finitely generated nilpotent groups M
is such a group the
= N
for all primes
M
with localizations
p . Mislin [9] had previously shown
that
G(N) is finite. Here an abelian group structure in
G(N)
is defined. If H
maps
: N -* H
primes with
and K
and \p : N •+ K
are two groups in G(N) then and families
P
P u Q = TT are constructed such that
equivalence and
ip is a
and Q of $
is a P-
Q-equivalence. The addition in
G(N)
is then defined by constructing the pushout square
N 4-
The group
L
is a representative of the sum in G(N) of the
genus class of H
and the genus class of K . The abelian
group defined in this way can for example be used to estimate the size of G(N) .
Stammbach: The work of Peter Hilton in algebra
44
Of course Peter Hilton's mathematical work continues. We certainly shall see in the future, as we have in the past, a lot of significant papers carrying that special trade mark: Peter Hilton. I have tried in this talk to describe some of Peter Hilton's work in algebra. However it is not possible to give in such a small space a real impression of the richness of his work and of the immense influence it has had on the mathematical community. There is another shortcoming too. I have said nothing about Peter Hilton as a person, nothing about his gentleness and helpfulness. To give just a glimpse of that aspect of his personality I would like to close this talk with a description of my first personal encounter with Peter. It was in spring of 1965? I was at that time a rather shy graduate student at ETH who had just obtained some first results on his PhD thesis topic. Peter Hilton who was spending a couple of weeks in Zurich, showed interest in my work and asked me to describe some of the details. We arranged to meet in front of the Forschungsinstitut at Zehnderweg. At the appropriate time I was there, waiting very nervously for my meeting with an eminent and important mathematician. Then Peter arrived. His first question was: Which language shall we use: english, french or german? Naturally I was relieved to be able to opt for german. His second question, in german of course, was: Haben Sie Zeit, m-itrnirdie London Times kaufen zu gehen? This is characteristic for Peter in more then one way. Not only is it evidence for the fact that Peter can hardly live without the daily issue of the London Times, but it also shows in a very subtle way Peter's thoughtfulness. No doubt, Peter had noticed my nervousness and thought that a walk and a little chit-chat would calm me down a bit. Indeed after our walk we had an intensive and satisfying talk on various mathematical matters.
Stammbach: The work of Peter Hilton in algebra
45
REFERENCES
For the references labeled H see the complete list of Hilton's publications in this volume. [1]
M. Auslander and I. Reiten, Representation theory of Artin algebras III. Almost split sequences, Comm. Algebra 3 (1975), 239-294.
[2]
L.S. Charlap and A.T. Vasquez, Characteristic classes for modules over groups I, Trans. Amer. Math. Soc. 137 (1969), 533-549.
[3]
B. Eckmann: Homotopie et dualite, Colloque de topologie algebrique, Louvain 1956, 41-53.
[4]
B. Eckmann and H. Kleisli, Algebraic homotopy groups and Frobenius algebras, 111. J. Math. 6 (1962), 533-552.
[5]
L. Evens, The Schur multiplier of a semi-direct product, 111. J. Math. 16 (1972), 166-181.
[6]
A. Gut, A ten-term exact sequence in the homology of a group extension, J. Pure Appl. Algebra 8 (1976).
[7]
A. Heller, The loop space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382-394.
[8]
A. Heller, Indecomposable representations and the loop-space operation, Proc. Amer. Math. Soc. 12 (1961), 640-643.
[9]
G. Mislin, Nilpotent groups with finite commutator subgroups. In: Localization in Group theory and Homotopy theory, Lecture Notes in Math. Vol. 418, Springer 1974, 103-120.
[10]
G. Mislin, Essay on Hilton's work in topology, this volume.
[11]
Y. Nomura, The Whitney join and its dual, Osaka J. Math. 7 (1970), 353-373.
THE DUAL WHITEHEAD THEOREMS J. P. May The University of Chicago
For Peter Hilton on his 6 0 t h birthday
Eckmann-Hilton duality has been around for quite some time and i s something we now a l l take for granted.
Nevertheless, i t is a guiding
principle to "the homotopical foundations of algebraic topology" that i s s t i l l seldom exploited as thoroughly as i t ought to be.
In 1971, I
noticed that the two theorems commonly referred to as Whitehead's theorem are in fact best viewed as dual to one another. details.
I've never published the
(They were to appear in a book whose t i t l e i s in quotes above
and which I contracted to deliver to the publishers in 19 74; 19 84, perhaps?)
This seems a splendid occasion to advertise the ideas.
The
reader i s referred to Hilton's own paper [2] for a h i s t o r i c a l survey and bibliography of Eckmann-Hilton duality.
We shall take up where he left
off. The theorems in question read as follows.
Theorem A.
A weak homotopy equivalence e :Y -• Z between CW complexes is a
homotopy equivalence. Theorem B.
An integral homology isomorphism e :Y •*• Z between simple spaces
is a weak homotopy equivalence. In both, we may as well assume that Y and Z are based and (path) connected and that e is a based map.
The hypothesis of Theorem A
(and conclusion of Theorem B) asserts that e^:ir^(Y) > TT^(Z) is an isomorphism.
The hypothesis of Theorem B asserts that e*:H*(X) -• H*(Y) i s
May: The dual Whitehead theorems an isomorphism.
47
A simple space is one whose fundamental group i s Abelian
and acts t r i v i a l l y on the higher homotopy groups.
Theorem B remains true
for nilpotent spaces, for which the fundamental group is nilpotent and acts nilpotently on the higher homotopy groups.
More general versions
have also been proven. It is well understood that Theorem A is elementary.
However,
the currently fashionable proof of Theorem B and i t s generalizations depends on use of the Serre spectral sequence.
We shall obtain a con-
siderable generalization of Theorem B by a s t r i c t word for word dualization of the simplest possible proof of Theorem A, and our arguments will also yield a generalized form of Theorem A. We shall work in the good category 0 of compactly generated weak Hausdorff based spaces.
Essentially the same arguments can be
carried out in other good topological categories, for example, in good categories of G-spaces, or spectra, or G-spectra.
An axiomatic setting
could be developed but would probably obscure the simplicity of the ideas. We shall use very l i t t l e beyond fibre and cofibre sequences. Let
XAY be the smash product X x Y/XvY and let F(X,Y) be the function
space of based maps X -• Y. The source of duality is the adjunction homeomorphism (1)
F(XAY,Z) 2 F(X,F(Y,Z)).
Let CX - XAl, LX = XAS 1 , PX = F(I,X), and fiX - FCS^X), where I has basepoint 1 in forming CX and 0 in forming PX.
For a based map f :X -• Y,
l e t Cf - Y <Jf CX be the cofibre of f and l e t Ff • X x PY be the fibre of f. X +• Y.
Let TT(X,Y) denote the pointed set of homotopy classes of based maps For spaces J and K, we have the long exact sequences of pointed
sets (and further structure as usual) (2) (3) The crux of Theorem A is the following t r i v i a l i t y ; we shall give the proof since nothing else requires any work.
May: The dual Whitehead theorems Lemma 1.
48
Let e :Y •• Z be a map such t h a t n ( J , F e ) - 0.
If hi± » eg
and
hiQ = f i in the following diagram, where iQ, i p and i are the evident i n c l u s i o n s , then t h e r e e x i s t
Proof.
g and h
which make the diagram commute.
Define k Q : J -• Fe by k Q ( j ) = ( g ( j ) , w Q ( j ) ) , where U>Q(J) t PZ i s
s p e c i f i e d by f(j,
l-2s)
if s < 1/2
h(j,
2s-l)
if s > 1/2 .
Choose a homotopy k :J A I + •>• Fe fromk.Q to the t r i v i a l map and define g:CJ -• Y and w : J A l + ^ PZ
Define
by
h:CJ A l + •• Z by h(j,s,t)
where u ( s , t ) - m i n ( s , 2 t ) and g and h
v ( s , t ) - max(y(l + t ) ( l - s ) , 2 t - l ) .
Then
make the diagram commute. We now introduce a general version of c e l l u l a r theory.
Definition 2.
Let ^ be any c o l l e c t i o n of spaces such t h a t I J e ^
if
J £ *1 • A map e :Y •*• Z i s said to be a weak Q»-equivalence if e*:Tt(J,Y) + I T ( J , Z ) i s a b i j e c t i o n for a l l J e ^f.
A 9*"" com P lex
is
a
space
X together with subspaces X^ and maps j n :J n ->• X^^ n ^ 0, such that Xo = {*}, J n i s a wedge of spaces i n ^ , X ^ .- C j n ' a n d the X n .
X is
the
union
The evident map from the cone on a wedge summand of Jn^i
i s c a l l e d an n - c e l l .
The r e s t r i c t i o n of j
n
of
into X
to a wedge summand i s c a l l e d
May: The dual Whitehead theorems an attaching map.
49
A subspace A of a ^-complex X i s said to be a
subcomplex if A is a ^-complex
such that
A^ Q XR and the composite of
each n-cell CJ •> An C A and the inclusion i:A •> X is an n-cell of X. Example 3.
Consider ^ = { Sn | n 2 0}, where we take Sn - IS 11 " 1 .
A weak
We call a ^ -complex X a cell complex.
If J n i s a
wedge of n-spheres, then X is a CW-complex with a single vertex and based attaching maps.
It i s easily verified that any connected CW complex i s
homotopy equivalent to one of t h i s form.
In general, the c e l l s of cell
complexes need not be attached only to cells of lower dimension. Other examples are of i n t e r e s t . set of T-local spheres
{I S In > 0},
For instance, (X might be the
where T is a set of primes.
In
t h i s case, ^-complexes lead to the appropriate theory of simply connected T-local CW-complexes. The acronym (due to Boardman) in the following theorem stands for "homotopy extension and l i f t i n g property". Theorem 4 (HELP).
Let A be a subcomplex of a ^-complex X and let e :Y •* Z
be a weak ^ - e q u i v a l e n c e .
If hi^ = eg and
diagram, then there exist
g and h
Proof.
nig = fi
in the following
which make the diagram commute.
By (1) and (3) and the fact that ^ Is closed under suspension, the
hypothesis implies that iT(J,Fe) = 0 for a l l J e ^ . compatible maps
g :X ->• Y and homotopies
We construct
h :X A I
^w
eg
-• Z from f I Xn to i**
by induction on n, starting with the trivial maps g
extending given maps
g
, and
h
, over cells in ^
e+j
and h
and
by use of the
given maps g and h and over cells of X n not in P^ by use of the case (CJ,J) already handled in Lemma 1.
50
May: The dual Whitehead theorems
In particular, taking e to be the identity map of Y, we see that the inclusion i : A + X i s a cofibration. Theorem 5.
For every weak CV -equivalence e :Y + Z and every ^-complex X,
e*:7r(X,Y) -•TT(X,Z) i s a b i s e c t i o n .
Proof.
We see that e* i s a surjection by application of HELP to the pair
(X,*). XA(3I)
It i s easy to check that X* I +
i s a Cl-complex which contains
as a subcomplex, and we see that e* i s an i n j e c t i o n by
application of HELP to t h i s pair. The c e l l u l a r Whitehead theorem i s a formal consequence. Theorem 6 (Whitehead).
Every weak ^--equivalence between ^-complexes i s
a homotopy equivalence. By Example 2, Theorem A i s an obvious special case. Now the fun begins.
We dualize everything i n s i g h t .
The dual
of Lemma 1 admits a dual proof which i s l e f t as an exercise. Lemma 1 .
Let e :Y •* Z be a mapsuch that Tr(Ce,K) • 0.
If p^h • ge and
PQII • pf in thefollowing diagram, where PQ, p^, and p are the evident projections, then there e x i s t
g and h which make the diagram commute.
•+ K
We next introduce the dual "cocellular theory." Definition 2 .
Let ^ be any c o l l e c t i o n of spaces such that ftK e JC i f
K e X • A map e :Y •> Z i s said to be a weak 7C -equivalence i f e*:ir(Z,K) + n(Y,K) i s a bisection for a l l K e ^ .
A K-tower i s a space X
together with maps X -• X n and k n lY^ *• 1^, n ^ 0, such that XQ - {*}, 1^
51
May: The dual Whitehead theorems i s a product of spaces in ) ( , X^^ - Fkn, and X i s the inverse limit of the X^ (via the given maps).
The evident map from X to the paths on a
factor of KQ_i i s called an n-cocell. called a coattaching map.
The projection of k
to a factor
is
A map p :X -• A i s said to be a projection onto a
quotient tower if A i s a X-tower, p i s the inverse limit of maps Xn -• A n, and the composite of p and each n-cocell A •>•A^ •• PK i s an n-cocell of X.
Example 3 . {0},
Let UL be any collection of Abelian groups which contains
for example the collection QAr of a l l Abelian groups.
Let %U'be
the collection of a l l Eilenberg-MacLane spaces K(A,n) such that A e (X and n ^ 0.
(We require Eilenberg-MacLane spaces to have the homotopy types
of CW-complexes; t h i s doesn't effect closure under loops by a theorem of Milnor.)
A KOJr -tower X such that 1^ i s a K(7Tn+1,n+2) for n £ 0 i s
called a simple Postnikov tower and s a t i s f i e s ffn(X) - iTn. map k
i s usually written k
Theorem 4
(coHELP).
n
I t s coattaching
and called a k-invariant.
Let A be a quotient tower of a )<.-tower X and l e t
e :Y -*• Z be a weak X-equivalence.
If p^h = ge and pgh = pf in the
following diagram, then there exist g and h
which make the diagram
commute.
Proof.
By (1) and (2) and the fact that
X i s closed under loops, the
hypothesis implies that 7T(Ce,K) = 0 for a l l K e X -
The conclusion
follows inductively by a cocell by cocell application of Lemma 1 • In p a r t i c u l a r , the projection p :X -»• A i s a fibration. Theorem 5 . e
For every weak X-equivalence e :Y -• Z and every 3C-tower X,
:TT(Z,X) > TT(Y,X) i s
Proof.
a
bijection.
The surjectivity and injectivity of e +
result by application of
coHELP to the quotient towers X -• * and F(I ,X) •> F(9I+ ,X), respectively.
May: The dual Whitehead theorems
52
The c o c e l l u l a r Whitehead theorem i s a formal consequence. Theorem 6 (Whitehead).
Every weak A-equivalence between ^C-towers i s a
homotopy equivalence. To derive useful conclusions from these theorems we have to use approximations of spaces by CW-complexes and by Postnikov towers.
For
a space X of the homotopy type of a CW-complex, we have Hn(X;A) = TT(X,K(A,n)). However, K -towers hardly ever have the homotopy types of CW-complexes. The best conceptual way around t h i s i s to pass from the homotopy category hJ
to the category
equivalences.
h J obtained from i t by i n v e r t i n g i t s weak homotopy
For any space X, t h e r e i s a CW-complex FX and a weak
homotopy equivalence y :I X -• X.
The morphisms of
h 3 can be specified by
[X,Y] = T.(rX,rY), with the evident composition.
By Theorem 5, we have [X,Y] = TT(X,Y) if X
has the homotopy type of a CW-complex.
Either as a matter of d e f i n i t i o n
or as a consequence of the fact t h a t cohomology i s an i n v a r i a n t of weak homotopy t y p e , we have Hn(X;A) = [X,K(A,n)] for any space X. Now r e t u r n t o Example 3 •
Say t h a t a map e :Y + Z i s an
C*.-cohomology isomorphism if e :H (Z;A) ->• H (Y ;A) i s an isomorphism for a l l A e CL •
If Y and Z a r e CW-complexes, then e i s an
ft.-cohomology
isomorphism if and only if i t i s a weak XCL-equivalence. Theorem 5 •
For every CL-cohomology isomorphism e :Y •> Z and every
XOL-tower X, e*:[Z,X] + [Y,X] i s a b i j e c t i o n . Proof.
We may as well assume t h a t Y and Z a r e CW-complexes, and the
r e s u l t i s then a s p e c i a l case of Theorem 5 •
May: The dual Whitehead theorems
53
This leads to the cohomological Whitehead Theorem, Theorem 6
(Whitehead).
e :Y -• Z in
The following statements are equivalent for a map
h-3 between connected spaces Y and Z of the weak homotopy type
of %ft--towers. (1)
e i s an isomorphism in
(2)
e^:7r^(Y) •»• TT*(Z) i s an isomorphism.
h3 .
(3)
e*:H*(Z;A) -»- H*(Y;A) i s an isomorphism for a l l
(4)
e*:[Z,X] -• [Y,X] is a bijection for a l l X&-towers X.
Aeft,
If 0- i s the collection of modules over a commutative ring R, then the following statement can be added to the l i s t . (5)
e*:H*(Y;R) •• H*(Z;R) i s an isomorphism.
Proof.
The previous theorem gives (3) ==> (4), (4) = > (1) i s formal, and
(1) <=*> (2) by the definition of h"3 ; (2) ==> (3) and (2) = > (5) since homology and cohomology are invariants of weak homotopy type, and (5) =**=> (3) by the universal coefficients spectral sequence. When ft- = Oil* , the implication (5) = > (2) i s the promised generalization of Theorem B.
It i s almost too general.
Given a space X,
i t i s hard to t e l l whether or not X has the weak homotopy type of a H. CUr-tower.
If X is simple, or nilpotent, the standard theory of
Postnikov towers shows that X does admit such an approximation. in Definition 2* with X = KGU^, each Kn can be an arbitrary
However, infinite
product of K(A,q) f s for varying q and the maps k n :Xn > K^ are completely unrestricted.
Thus /Cft^-towers are a great deal more general than
nilpotent Postnikov towers; compare Dror [1]. The applicability of Theorem 6 to general collections (K i s of considerable practical value.
A space X i s said to be C^-complete if
e :[Z,X] •• [Y,X] is a bijection for a l l U.-cohomology isomorphisms e :Y -• Z.
Thus Theorem 5 a s s e r t s that 7C(K-towers are
fl-complete.
The
completion of a space X at CL i s an fl--cohomology isomorphism from X to an ft.-complete space.
For a set of primes T, the completion of X at the
collection of T-local or T-complete Abelian groups i s the localization or completion of X at T; in the l a t t e r case, we may equally well use the collection of those Abelian groups which are vector spaces over Z/pZ for some prime p e T.
May: The dual White he ad theorems
54
These ideas give the starting point for an elementary homotopical account of the theory of localizations and completions in which the l a t t e r presents l i t t l e more difficulty than the former; compare Hilton, Mislin, and Roitberg [3].
Details should appear eventually in
"The homotopical foundations of algebraic topology".
The equivariant
generalization of the basic constructions and characterizations has already been published [4,5], and there the present focus on cohomology rather than homology plays a mathematically essential role.
Bibliography 1. E; Dror. A generalization of the Whitehead theorem. Lecture Notes in Mathematics Vol. 249. Springer (1971), 13-22. 2. P. Hilton. Duality in homotopy theory: a rectrospective essay. J. Pure and Applied Algebra 19(1980), 159-169. 3. P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups and spaces. North-Holland Math. Studies Vol. 15. North Holland. 1975. 4. J. P. May, J. McClure, and G. Triantafillou. Equivariant localization. Bull. London Math. Soc. 14(1982), 223-230. 5. J. P. May. Equivariant completion. Bull. London Math. Soc. 14(1982), 231-237.
HOMOTOPY COCOMPLETE CLASSES OF SPACES AND THE REALIZATION OF THE SINGULAR COMPLEX D.Puppe Mathematisches Institut der Universitat Heidelberg D-69OO Heidelberg Federal Republic of Germany Dedicated to PETER JOHN HILTON on the occasion of his 6oth birthday We give a new proof for the well known theorem that the canonical map
from the geometric realization of the singular complex of a space X is a homotopy equivalence if X has the homotopy type of a CW-complex. The theorem is due to Giever EG, 6.Theorem VI] and Milnor [Mi, 4.Theorem 4 ] . Proofs of this or closely related theorems may also be found in [G-Z, VII ], [L, VII.10.10] and [May, 16.6]. Our proof is elementary in the sense that we use only some basic geometric constructions (like the double mapping cylinder and barycentric subdivision) and e.g. do not need homotopy groups or homology. For this purpose we study classes of (topological) spaces which are komotopy cocompl&te., i.e. closed under all homotopy colimits, and we show in particular: (A) Tkt da** E ofa t>pacbi> X &OK wklck z *A a komotopy zquuvaJLo-nca u> komotopy co complete.. (B) Ike, cla*6 W o& ApaceA kavlng tke. komotopy type, o£ a CW-complzx AJ> contained In any komotopy co complete. C1JXM> u)klck contains a one, point 6pac&. Obviously (A) and (B) imply the Giever-Milnor theorem on e and also the fact that W is the smallest homotopy cocomplete class which contains a one point space. This latter fact can, of course, also be seen directly. Some simple categorical arguments give the corollary that £_, x is a weak homotopy equivalence for
Puppe: Homotopy cocomplete classes
any space
X (Section 3 ) . The author wishes to express his thanks to the Mathematical Institute of the National University of Mexico, where he was a guest when this work originated. 1. Homotopy cocomplete classes of spaces Top of all We can work either in the category topological spaces or in some other category suitable for homotopy theory like compactly generated spaces. What one really needs about the underlying category is very little and can easily be abstracted from the following. For simplicity we talk about Top 1.1. DEFINITION. Let C be a class of objects of Top (which we sometimes identify with the full subcategory of Top having those objects). C is called homotopy cocomploXd if the following three conditions are satisfied: (a) li X € C and X1 h
In
C
AJ> an oJLzmznt o^
Recall t h a t
Z(f-,f2)
C .
i s o b t a i n e d from t h e sum
X- U (X x i ) U Xo 1
O
Z
by the identifications (x,0) ~ fxx
for all
x € XQ . There is a general notion of homotopy colimit for any (small) diagram in Top [V, (1.1)]. It is not hard to show that C satisfies (a) - (c) if and only if it is closed under all such homotopy colimits; but we shall not use this in the present paper.
56
Puppe: Homotopy cocomplete classes 1.2.
57
EXAMPLES. The f o l l o w i n g c l a s s e s o f s p a c e s a r e homotopy
cocomplete: (i)
The class
{0} .
(ii)
The class W of spaces having the homotopy type of a CW-complex. (iii) The class hi of spaces which are numerably locally contractible. This is trivial for (i). For (iii) the necessary definitions and proofs may be found in [P, Section 1] where it is also observed that the class W is strictly larger than W . About (ii) the following three things may be said: 1. The assertion is well known. 2. To check it one needs only to observe that conditions (a) and (b) are trivial for W , whereas (c) follows from the "homotopy invariance" of the double mapping cylinder by "cellular approximation". 3. It will also be an immediate corollary of 1.3 and 2.1 below. 1.3.
PROPOSITION.
cocomplatz and contain* contain*
Lvt C be a oXa&h ofi i>pac which. ik homotopy a one point Apace. Then
W <= C , i . e. C
all 6pacei> ofa the homotopy typo, o^ a CW-complex.
Observe first that each one point space pt belongs to C (by (a)) and hence each discrete space (by (b)). In particular S° € C . Using (c) we get Sn € C for all n by induction because S n is the double mapping cylinder of
Now let A be any CW-complex and denote by A its n-dimensional skeleton. Then A° € C because it is a discrete space. For any n i 1 one obtains A n from A "" by attaching n-cells which can be described by saying that A is the double mapping cylinder of
where
D is a suitable discrete space and
f some
Puppe: Homotopy cocomplete c l a s s e s
58
c o n t i n u o u s map. Using (c) we g e t A £ C f o r a l l n byi n d u c t i o n , b e c a u s e we know a l r e a d y t h a t D and n S - x D (which is the sum of copies of S n s ) are in
c.
Now
A
is the colimit of the sequence
A° cz A 1 e A 2 c: . . . and because each inclusion map is a cofibration the colimit A is homotopy equivalent to the telescope (= homotopy colimit) A of the same sequence [Dl, 2.Lemma 6 and Remark 1]. But A may be described as the double mapping cylinder of f^ n n odd where
n
->
11 A
n even
f-
maps the summand A into A n by the identity if n is odd An by inclusion if n is even and f does the same with "odd" and "even" interchanged. By (b) all the spaces in the above diagram are in C . Hence A € C by (c) and A E C by (a). Since A was an arbitrary CW-complex we have W <= C (using (a) once more) . 2. The realization of the singular complex Let X be a space. By SX we denote the "singular complex" of X considered as a simplicial set, i.e. SX consists of the sets of singular simplices
together with face and degeneracy operators. If K is any simplical set then |K| denotes its geometric realization in the sense of Milnor [Mi]. It is obtained from (K x A n ) n
Puppe: Homotopy cocomplete c l a s s e s
59
by the well known i d e n t i f i c a t i o n s using both faces and degeneracies. For each space ex:
|SX|
X one has a canonical map
> X
which is induced by sending € SnX x An
(a,t) into 2.1.
a(t) € X . THEOREM. The clan
homotopy equivalence.
E o& Apace*
ii> homotopy
X £o* which
e x U> a
cocomplete.
If X is a one point space so is ISX| . Hence E contains all one point spaces. Combining 1.3 with 2.1 we get (d c E . On the other hand |sx| is always a CW-complex, which obviously implies E c W . Hence we have W = E and we may state 2 . 2 . COROLLARY. The following thn.ee CIOAAQA o£ Apace* axe the A me: (7) The CIOAA oft ApaceA having the homotopy type o^ a CW-complex. (2) The CIOAA ofi all x fan which e : |sx| > X Ja a homotopy equivalence. (3) The hmallebt clahh o^ Apacet> which iA homotopy cocomplete and contain* a one point Apace. In the rest of this section we give the proof of Theorem 2.1. We have to verify conditions (a), (b) and (c) of Definition 1.1 for the class C = E . Condition (b) is trivially satisfied because the functors singular complex X I > SX and geometric realization K i * |K| both commute with arbitrary coproducts. Condition (a) is not much harder. It follows from the fact that both functors preserve homotopies. To say this in more detail let A[n] be the standard simplicial set with A[n]
= {(weakly) increasing maps [q] [
> [n]}
Puppe: Homotopy cocomplete classes
where
[n] = {0,1,..., n} . Then
60
|A[n]| = A n .
In fact we need only the case n = 1. For any simplicial set
K
one has the canonical
map (of simplicial sets)
nR : K sending
x € K
which maps
>
S|K
into the singular simplex
t € A
a: A
into the equivalence class of
(x,t) . In particular this gives A[l]
* SA 1
which we use to form |SX| x A 1 = |SX| x |A[1]| = |SX x A[1]| lid*nl , icy x
Let
cp denote this composition. N o w if h: X x A 1
*Y
is a homotopy connecting
f,g : X
>Y
t o each other,
then |Sh] - cp : | S X | x A 1 is a homotopy connecting
|sf| t o
> IsCXxA 1 ) | |Sg|
> | SY |
.
This shows that t h e composite functor X i > ISX| preserves h o m o t o p i e s . (We need n o t talk explicitly about simplicial homotopies.) It follows that if f: X > X 1 is a homotopy equivalence so is |Sf|, a n d t h e diagram
Isx '*! x
Isfl
» Isx 1 > x1
shows that condition (a) holds. We now turn to condition (c) which is,
Puppe: Homotopy cocomplete classes
61
of course, the crux of the matter. Let f X
f
l X
l "
2 >X2
o
be given. The double mapping cylinder abbreviate by X
Let
Y-
which we
Y is a quotient of
1U
(X
O
X T) U X
2 V
be the image of 1
in
Z(f , f j
Y , and
o Y2
4
the image of
( X Q x [1,1], u X 2 . Finally let Y o = Y x n Y 2 • T h e n Y o " X o * ["4'"4] a n d there are obvious canonical homotopy equivalences i = 0, 1, 2 . Now let us make the hypothesis that for i = 0,1/2. This implies already that
E
is closed under homotopy equivalences
(condition (a)). We have to show that ey:
X. £ E
Y. £ E , because we know Y £ E , i.e. that
|SY
is a homotopy equivalence. Instead of doing this directly we consider a different kind of geometric realization. If
K
is a simplicial set we denote by ||K|| the space ob-
tained from i
(K x A n ) n
by making only those identifications which correspond to face operators, i.e. for each strictly increasing map a:
[m]
* [n]
(a*x,t) with
and all
x £ K^ ,
(x,at) . (In the case
t £ Am K = SX
back to Giever [G].) °ne has a canonical map
we identify this goes
Puppe: Homotopy cocomplete classes
which is known to be a homotopy equivalence [D2, Proposition 1]. Composition with e gives a canonical map
for each space and it makes no difference whether is a homotopy equivalence. we say that e or that We are going to show that £y i s a homotopy equivalence. For this we consider the diagram
IsiJ ISY^I *
ljsi2! ||SY J | E
Y
The maps i., i~ are inclusions. All horizontal maps are cofibrations, and all vertical maps are homotopy equivalences. By the glueing lemma [B, 7.5.7] or [Dl, 2.Lemma l] one obtains a homotopy equivalence e ' between the pushouts of the two lines. The pushout of the lower line is obviously Y . The pushout of the upper line is the realization ||SY1 U SY2J| of the sub-(simplicial set) SY± U SY. of SY consisting of those singular simplices of Y which lie entirely in or entirely in We have a commutative diagram
Thus in order to prove that £' is a homotopy equivalence it suffices to show that ||SY1 U SY21| is a deformation retract of llSY)i . This is a special case of
62
Puppe: Homotopy cocomplete classes
2.3.
LEMMA. that
tkt
Lzt
Y be a Apace, and
intojiloKA
U a &ub*t&t> o£ Y
o£ tkd dL2.mz.wtA o^
S(Y;U) = U
veu
SU
63
(J COVQA Y . Let
cz SY .
Tke.n ]|s(Y;U)|| u> a Ptwofa. In the standard setup of singular homology one proves the excision theorem by showing that cz S(Y;(i) > SY induces a homology isomorphism. What we do here is just translate this proof (whose main tool is barycentric subdivision) into homotopy. Barycentric subdivision for us will be a certain map
defined (for any space X ) as follows: If [c,t] is the point of ||sx|j represented by (a,t) € S X x A n then we choose an affine (injective) simplex u: A q >A belonging to the ordinary barycentric subdivision of A such that t is in the image of u. (More precisely: There is a strictly ascending sequence of q+1 faces of A n such that u sends the vertices of A q into the barycenters of these faces preserving the order.) Then we define b x fa,t] = [au / u~ 1 t] . It is easy to check that this neither depends on the choice of u nor on the choice of the representative (a,t) of [a,t] . (For the latter one uses the compatibility of barycentric subdivision with injective simplicial maps. Such a compatibility does not hold for "degeneracy maps" A »A / P < q / and this is the reason why we use the modified realization )| |j at this point and not | | .)
Puppe: Homotopy cocomplete c l a s s e s
64
We also need a homotopy
from the identity of HSX(\ to b
. This is defined in a
x
similar way as bx : We consider the simplicial subdivision of A x I whose vertices are the vertices of A n x o and the barycenters of faces of A n x 1 . More precisely: Let e ,...,e be the vertices of A and let b. . be the barycenter of the face of A 1
• • •1
o q spanned by e. ,.-.,e. . Then we take the set x 1 o q V = {(e±/0)|i = 0,...,n} U { ( b
i
o
ir..i
>1]\°
and ( ge .i v, 0e ) i t< (teh^e Of o) l l o wi fi n g pi a
( e
i
i
o"
1)
•V
,0)
i
f l )
q
if
*1 < ' • • \
*
n }
order
if
< (b .
<
* ^
* <-
{i
o
y
••• /
o •
The totally ordered subsets of V form our simplicial subdivision of A x I . Now if (a,t) € s X x A n and s € I then we n q choose a simplex u: A » A x j belonging to the subdivision just defined (i.e. an affine map whose restriction to the set of vertices is a strictly increasing map into V ) and such that (t,s) is in the image of u . We define hv([a,t],s) = [airu/u"1(t,s) ] ,
x
where IT: A x I »A is the projection. Again it is easy to check that this neither depends on the choice of u nor on the choice of the representative (o,t) of [a,t] , and it is also easy to check that h is
x
Puppe: Homotopy cocomplete classes
65
indeed a homotopy starting at the identity of || SX )| and ending at b . It is clear that b_x_ and hxv are both natural with respect to continuous maps of X . In particular, returning to the hypotheses of the lemma, they are natural with respect to the inclusions U <= y , u € U . Hence b y is a map of the pair ( ||SY|| , |[S (Y; U)j[ ) into itself and h is a homotopy of such maps. If we fix for the moment one singular simplex a: A q > Y , then by compactness and standard arguments about the size of simplices in the barycentric subdivision of A q there is a natural number k such that each simplex of the k~th barycentric subdivision of A q lies entirely in a" U for some U £ U . This means that the k-th iterate of b maps [a,t] into q ||S(Y;U)i| for each t € A . Since ||SY|( is a CW-complex (canonically) and ||s(Y;U)l| is a subcomplex the lemma will be proved if we show that for each n any map of pairs
( D the n-disc) is homotopic (as a map of pairs) to a map which sends all of D into the smaller space ||S(Y;U)|| (cf. e.g. [W, II. 3.1 and 3.12]). By compactness f(Dn) is contained in the union of finitely many closed cells of ||SY|| which are nothing else but sets of the form {[a,t]|t e A q }
,
a: A q
*x
fixed.
As we just remarked there is a natural number k such that (b ) maps each of these cells and hence f(D ) into ||S(Y;(J)H . Since b is homotopic to the identity of ( JSY||, ||S(V;U)II ) as a map of pairs so is the iterate k k (b ) . Hence f is homotopic to (b ) f as a map of pairs, and the latter sends all of D n into ||S(Y;U)|| . This proves the lemma.
Puppe: Homotopy cocomplete classes 3.
66
Concluding remarks
From the r e s u l t s of Section 2 one obtains 3.1.
COROLLARY. Tkz map e : |sx|
fan, any 4pace,
^X
AJ> a w&ak
easily
komotopy
X
X.
This follows from 2.2 and t h e following p u r e l y
categorical
observation. 3 . 2 . PROPOSITION.
category
iat
(T,6,e)
be a comonad OVQA i>omz
V . Thin fan alt obj&cti> A,X o£ V thz map
AJ> Ausijactive,.
It Ltb injactLvz
i,^ both
e
and e
a/ie.
oft 3.1* Let P be the homotopy category of topological spaces and (T,5,e) the comonad associated to the pair of adjoint functors K! »|K| , X I-—> SX ,
K a simplicial set X a space
[Mac, VI]. Then TX = |sx| . (Strictly speaking we get first a comonad on Top , but it induces a comonad on the homotopy category because T preserves homotopies, cf. 2.1 proof of condition (a).) The counit of the comonad is just (the homotopy class of) the map e : |SX| »X considered above (whereas we do not care x what the diagonal is in our concrete situation). Thus we know from 2.2 that e is an isomorphism in V for all spaces A (because TA € W ) . Hence 3.2 implies that r induces a bijection of homotopy sets £XJ|C:
for all spaces we may replace
[TA, | SX | ]
> [TA,X]
A,X . If A € 0/ then TA by A .
e : TA = A and
& 3.2. One of the axioms of comonads says that
Puppe: Homotopy cocomplete classes
commutes for all objects triangle (for
A
X
of
instead of
67
V . Using the left hand
X ) and the fact that
e
is
a natural transformation one verifies that the composition 6 £ T 2 A X* V (TA, X) —=-» V (T A, TX) — ^ - » P (TA, TX) — » V (TA, X)
is the identity which proves the surjectivity of £
x* • A s s u m e now that
erriv
the above diagram shows that
, hence
e
X
= Te J. X
6
and
Te
Then
are also
is inverse to both e___ and 6V x ix . W e claim that the triangle
isomorphisms and that Te
is an isomorphism.
X.
P(TA,TX)
c o m m u t e s . To see this let
T2A £
^ - >
f € P(TA,TX)
T2X
| £ TX
TA| TA
. Then
= T£x
> TX
and we have indeed £
If also and hence
$Af =
eT7v ev
f£
TA=
^x
) ( T f )
=T(£Xf)
is an isomorphism then
ej,
= T £
X*
f
*
is bijective
injective.
3.3. REMARK. Traditionally (except in [G]), before looking at the counit
Puppe: Homotopy cocomplete c l a s s e s ex:
|SX|
68
>X
of the adjoint pair Kl
>\K\ ,
K
X 1
> SX ,
X
a simplicial set a space
one looks at the unit
nK: K » S | K | and p r o v e s t h a t i t i s a (weak) homotopy e q u i v a l e n c e of s i m p l i c i a l s e t s . We g e t t h i s r e s u l t from 2 . 2 and t h e c o m m u t a t i v i t y of
REFERENCES
[B]
R.BROWN, Elements of modern t o p o l o g y , McGraw-Hill London 196 8. [Dl] T.tom DIECK, P a r t i t i o n s of u n i t y i n homotopy t h e o r y , CompoA.Matk. 23 (1971), 159-167. [D2] T.tom DIECK, On t h e homotopy t y p e of c l a s s i f y i n g s p a c e s , manuAc/iipta math. 11 (1974), 4 1 - 4 9 . [G] J.B.GIEVER, On t h e e q u i v a l e n c e of two s i n g u l a r homology t h e o r i e s , Ann. o£ Math. 51 (1950), 1 7 8 - 1 9 1 . [G-Z] P.GABRIEL and M. ZISMAN, CalculuA o£fisiactLonAand komotopy thzoKy, E r g e b n i s s e d e r Mathematik und i h r e r G r e n z g e b i e t e 3 5 , S p r i n g e r 196 7. [L]
K.LAMOTKE,
Smi&impLLzAjodLz alg
Die
Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 147, Springer 1968. [Mac] S.MAC LANE, Cat&gonizA ion tkz wo/iklng mcUkmatidian, Graduate Texts in Mathematics 5, Springer 19 71. [May] J.P.MAY, Slmptidal objzct6 In aigzb/uUc topology, van Nostrand Mathematical Studies 11, van Nostrand 1967.
Puppe: Homotopy cocomplete classes
69
[Mi] J.MILNOR, The g e o m e t r i c r e a l i z a t i o n of a s e m i s i m p l i c i a l complex, Ann. ofi Ucutk. 65 (1957), 357-362. [P] D.PUPPE, Some w e l l known weak homotopy e q u i v a l e n c e s a r e g e n u i n e homotopy e q u i v a l e n c e s , Symposia MathmouLLca 5 (1971), 363-374. [V]
R.M.VOGT, Homotopy limits and colimits, (1973),
[W]
Mcutk.Z. 134
11-52.
G.W.WHITEHEAD,
EIQJYKLWU ofa komotopy thdony,
Texts in Mathematics 61, Springer 1978.
Graduate
BETTI NUMBERS OF HILBERT MODULAR VARIETIES E. Thomas Department of Mathematics University of California Berkeley, CA 9*+720 A.T. Vasquez Graduate School, CUNY 33 West i+2nd Street New York, NY 10036 1
INTRODUCTION In this paper we give formulae for the Betti numbers of
Hilbert modular varieties, as -well as show that any such variety is simply-connected.
These results complement the work of [19], where we
calculate the Chern numbers of modular varieties of complex dimension three.
The goal is the classification of modular varieties up to diffeo-
morphism, birational equivalence, or biholomorphic isomorphism; see section three for further discussion. Throughout the paper number field of degree (= PSLp(O))
n
K
subgroup of
G
subgroup (or
if either
of finite index.
We will say that a group
T = G
or
T
T
V
is a principal congruence
T
is of principal type.
G
acts on
H
embeddings of
K
(the complex upper half plane) by Thus, by means of the
n
in the real numbers, we obtain an action of
V ) on
H
n
. We define
is
is a torsion free
If
The group
Y-p
G
T = G ) , we say that
linear fractional transformations.
action.
its ring of integers, and
its Hilbert modular group.
of modular type (for K )
hence of
will denote a totally real algebraic
(>l) , 0
distinct G
(and
Y r = iP/T , the orbit space of this
is a non-compact normal complex space with a finite number
of isolated (''quotient") singularities, the images of the elliptic fixed points of the action of Let F = G , h Yp
Hn .
,h denote the number of parabolic orbits of K . ) By adjoining
T . (if h
points
Y-p one obtains a compact protective algebraic
with isolated singularities: the cusps and the quotient
singularities.
See [ll] for a detailed treatment of this material.
call any non-singular model
r.
on
is simply the class number of
(called "cusps") to variety
T
Z_, of
Yp
We
a Hilbert modular variety for
Thomas & Vasquez: Betti numbers of varieties
71
Our first result is: THEOREM 1.
Let
group of principal type and
K
be a totally real number field,
Z^
a modular variety for
V .
V
a
Then,
Z_
is
simply-connected* This has been proved for quadratic number fields by Svarcman [l6].
Moreover, he shows that for all
n
the singular varieties
?„
are
simply-connected. Note that by the theorem
b (Zp ) = 0 .
•f- Vi
the
i
»L
Betti number of a space
X ) .
(Here
1
b.(x)
denotes
1
In the following section we give
(computable) formulae for all Betti numbers of modular varieties, e.g., see Tables 3.2 and 3.^. If b^ ,
deg K = 3
9
we have an especially simple formula for
at least for certain modular varieties.
obtained from
Y^
Recall that
Z_
is
by resolving the cusp and quotient singularities.
Ehlers [3] gives an explicit method for resolving cusp singularities, while in [19] we give resolutions for the seven types of quotient singularities that arise for cubic number fields. modular type for a cubic number field variety
Z^
If
V
is a group of
K , we will say that a modular
is special if: (i)
the cusp singularities in
Y^
are resolved using
Ehlers1 construction; (ii)
the quotient singularities of
Yy
are resolved using
the models given in §2 of [19]. We then have: THEOREM 2.
Let
K
be a totally real cubic number field,
a group of modular type for
K
and
for
a special Hilbert modular variety
T . Then, b 3 (z r )
Here
Z^
=
8(1 - x(r))
.
x{Y) = x( z p) = arithmetic genus of
Z^ .
genus is a birational invariant (see §U), x(^)
(Since the arithmetic depends only on
Using a result of Freitag we may write way.
T
For each positive integer
i
set
b^
T . )
in a different
Thomas & Vasquez: Betti numbers of varieties
72
S.(F) = dimension of the complex vector space of cusp forms (for
r ) of weight
Freitag shows (7.2 in [k]):
2i .
= (-l)n(x(r) - l) , where
S
deg K = n .
Thus, by Theorem 2, we obtain: COROLLARY 1.
^(Zp) =
8S ( D
.
In the next section we give some evidence for the following conjecture (beyond the fact that
8= 2
! ) .
CONJECTURE 1. Let K be a totally real algebraic number field of odd degree
n9
with
V a group of modular type for K . Then
there is a Eilbert modular variety •U
( *7 \
n 2
—
( V \
O^O
F
Zp such that:
1
BETTI NUMBERS
In this section we give formulae for the Betti numbers of modular varieties.
Suppose that
quotient singularities; let
Yp has
E. ,
h
1 < i
cusp singularities and s and
Q. ,
denote resolutions of these respective singularities. regard the
E!s
z Thus,
and
Qfs
Yp is a manifold with boundary, namely
,
i
Topologically, we
as manifolds with boundary, and write:
= xi u E u . •. u E
E =1 E
1< j < s ,
u Q u ••• u Q 3(Q.) J!
J « 1 0. . j
J
Since t h e E f s and Q?s a r e a l l d i s j o i n t we have for k ^ 1 , k
i
k
i
k
3
k
j
We prove: THEOREM 3. Let K be a totally real algebraic number field of degree
n _, T a group of modular type for K and Zp a Eilbert
modular variety for T . Then, for
l ^ k ^ n - l .
Thomas & Vasquez: Betti numbers of varieties
73
VV = where
The proof relies on work of Harder and is given at the end of the section. Suppose now that
x is a cusp singularity in Y_ . As shown
in [ll], x is characterized by a pair Z-module of rank of rank
(M, V) , where
E of x "by Ehlers1 method,
n - 1 . To construct a resolution
one starts with a "complex of simplicies" M
M is a free
n in K and V is a group of totally positive units
and V . The group
£ , defined using data from
V then acts "simplicially" on £ . (For details,
see §1 in [3]; also, §1 in [l8] for a brief resume of the method.) For 1 ^ i ^ n , we set
where
Z
denotes the set of i-simplices in
from [3] that
*>2n_2(E) = N^E) ,
and so
£ .
I t follows readily
^ 2 n _ 2 (E) = ^ ( E ) = £ N ^ E j .
Thus, as a corollary t o Theorem 3 we obtain (if the
E. ! s
are constructed
by Ehlers' method):
W
=
V a ) + Ni(E) +n •
(2 1)
-
In the following section we give some concrete examples. Since
Zy has real dimension
2n , by Theorem 3 and Poincare
duality we are left with only the calculation of b (Z^) . Of course, by definition of the Euler characteristic, b (Zr) = n
(-l)n+1[2 1
However, we wish to express the field
i=1
b
K , as in Theorem 3.
e , we have
( - D V t Z j - e(Zr) + 2] . I
simply in terms of data from
(2.2)
E, £ and
For this we have:
THEOREM k. Let K be a totally algebraic number field of degree
n j with
Z^ a Hilbert modular variety. Then,
Thomas & Vasquez: Betti numbers of varieties bn(Zr) Here
A(n) =
( - l ) n + 1 ( 2 • A(n) - e ( Y r ) ) + b (fi) + b (E)
=
n-1 I (-l)16(n, i=0
e(Y r ) I
i)
.
Thus we have expressed
b
in t h e d e s i r e d
Proof of Theorem 4. dimensional b o u n d a r i e s , e(Yp) + s ,
since
Moreover, by Hirzebruch
2[G : r ] £ _ ( - l ) + uI a ( T ) r " K r r
=
74
Since t h e
X
.
[ll],
.
fashion. E.fs
and
Q's
e ( Z r ) = e(Q) + e(E) + e(Yp) .
have odd-
But
e(Y ) =
Using (2.2) and t h e fact t h a t F e a c h
e(disk) = 1 .
i s a l e n s space, Theorem k now follows from Theorem 3 and t h e
8Q
following
fact.
r
by Ehlers
FACT. Let (E, 3E) denote any cusp resolution method. Then, e(E) =
where
2
^
constructed
J
b. = b. (E) . We give the proof at the end of the section. Suppose now that
n
is an odd integer
2 n ~ 1 ; also, Vigneras [21] has shown that
( > l ) , then
A(n) =
e(Y r ) = 2 n x(T) . Thus by
Theorem k we have: COROLLAEY 2. Suppose that *n(zr)
=
K
has odd degree
2 n s 1 ( D + b n (£) + b n (E)
n , Theny
.
Notice that by the corollary, Conjecture 1 follows from: CONJECTURE 2. Suppose that quotient singularity resolution (i) bn(Q) and each cusp resolution (ii) bn(E)
= E =
0
Q
K
has odd degree
can be chosen so that
;
can be chosen so that 0
,
n .
Then each
Thomas & Vasquez: Betti numbers of varieties Proof of Theorem 2. Suppose that resolution
Q
75
deg K = 3 and that the
is chosen to "be one of the seven resolutions constructed
in §2 of [19] • Then, by Theorem 3.^ of [19], condition (i) given in Conjecture 2 is satisfied by LEMMA. 1. Let
Q . Thus, Theorem 2 follows at once from
K be a totally real aubio number field and
a resolution of a eusp singularity for
E
K constructed using an Ehlers '
3-complex. Then, b (E) = 0 . We give the proof at the end of the section. Proof of Theorem 3. We set
Z° = Y° U £ Since each
8Q
so that
dZ°v = 3E .
is a lens space (and hence, rationally, a sphere) we see
that
b ± (zj) = b ± (Yj) + b.((Z) \ Suppose now that
E
for
i > 0 .
is a resolution of a cusp singularity, with
(2.3) 9E as
a boundary. Taking homology with complex coefficients we have: CLAIM. For
1 < i < n - 1 , H.(3E)
•H.(E) is injective.
Assuming this for the moment we have: Proof of Theorem 3. By the Claim, the following two sequences are exact, for
1 < i < n - 1 : >E±(E9ZE)
(a) 0
•H.OE)
•H.(E)
(b)
• E±(dE)
• H^zJ!) « E±(E)
0
T h e r e f o r e , by (a) and ( b ) , b (z }
i r
=
b (z ) + b
i ?
i(E) " bi(3E)
= ^(zj) + b ^ E , 3E) . Consequently, by Lefschetz duality and (2.3) we have:
• H ± (Z r )
Thomas & Vasquez: Betti numbers of varieties
76
(c) D±(Zr) = ^(yj) + \>±(Q) + b ^ ^ C E ) . Denote by
S t h e s e t of quotient s i n g u l a r i t i e s in
t h e homotopy type of ^(Yp)
Y^ - S ;
Y_ .
Thus,
Y
has
moreover,
= b ± (Y r - S)
= 'b i (Y r )
,
0 < i < 2 n - 2
.
Thus t h e proof of Theorem 3 follows a t once from ( c ) , when we show: (d)
For
K i < n - 1
Suppose first that
T
b
,
j/V
=
6 ( n
is torsion free.
i )
'
•
Then (d) follows
from the work of Harder [8] (see also [5]). By Theorem 2.l(i), Remark (ii) on page 1^-5, and the reference to the work of Matsushima and Shimura given on page ikj (all references are to [8]), we see that H 1 (Y r )
=
H^(Y r )
,
1 < i
complex coefficients
where the right hand group is isomorphic to the elements of degree the exterior algebra generated by certain 2-forms on (op.
cit., page lU6) •
Thus,
i in
Y^ , rj , ... , ri
dim H.(Yp) = 6(n, i) , which proves (d)
and hence Theorem 3, in this case. On the other hand, suppose that F = G . • Let F'
Ff
T
has torsion; i.e.,
be a torsion free subgroup of finite index —
to be an appropriate principal congruence subgroup.
so that
F
acts on Yp, , with
e.g., take
Set F = F/FT ,
Y = Ypf/F . Thus, by Theorem III.2.U of
[8], H*(Y r )
«
H*(Yp)F
=
submodule of H*(Yr,) by
H 1 (Y r ,) « H^(Ypt ) »
As observed above, ^(Yp) ^ H ^ ( Y r J
F
.
fo
PSLp(R) -invariant.
Hence, each
F-invariant.
H^(Y r? ) F = H^(Ypt) ,
6(n, i) ,
as before.
1 < i
r|.
and so
n]L, . . . , nn
where they are in fact
is G-invariant and so (on which implies
Y^f ) ,
dim HX(Yr) =
This completes the proof of the theorem.
Proof of Claim, result:
^
But Harder [8] shows that the 2-forms
(see above) are a l l defined back on H , Thus,
left invariant
F .
It clearly suffices to prove the analogous
Thomas & Vasquez: Betti numbers of varieties
(e) For As shown in [ l l ] , I11"1 .
9E
fibers over the
By Proposition 1.1 in [ 8 ] ,
1 < i
H1(E)
1
Since
H*(T ~ )
n - 1
77
• HX(3E)
t o r u s , say
is onto.
p : 3E
p* : H^T31"1) « H^SE)
,
•
for
i s generated "by 1-dimensional classes,
i t suffices t o prove (e) simply in the case
i = 1 .
For t h i s we need
the following r e s u l t , which will be proved in section 5. LEMMA 2 . Thus
b (E) =
H (E)
n - 1
and H (3E)
.
are both complex vector spaces of
the same dimension, and so to prove (e) (with
i = 1 ) , it suffices
to show that
This follows by exactness
H (E)
if we know that Hp
• H (3E)
is injective.
H ^ E , 3E) = 0 . But by Lefschetz duality,
.(E) , which is zero since
H 1 (E, 3E) *
E has the homotopy type of the singular
2n - 2 ) . This completes the proof of the Claim.
set (of dimension
Proof of Faot,
By the exactness of sequence (a) above, and by
Lefschetz duality,
where
c. = b.(3E) . Thus,
e(E) = T (-l)V + (-D\ j=0
J
n-1
n
.
n-1
= 2 - 1 (-DJb?ri , + (-1)% + I n J n j=0
"
j=0
But as shown above, in the proof of the Claim, c_ 1 where
Tn"*
= b. (T n ~ ) l
,
1< i < n -1 ,
is an (n - l)-torus. n—1
.
Thus,
T
This completes the proof. Proof of Lemma 1*
We use the notation from [3] and [IT].
Thus, we assume that the resolution with
E
is given by a pair
(M, V) ,
E = X/V for a certain (open) complex n-manifold on which the group
Thomas & Vasquez: Betti numbers of varieties V
acts freely and properly dis continuously. V?
find an appropriate subgroup N (E f )
(i)
Let that each R
R Let (x^
I ,
while
I : C
*R
X2. x )
We regard
T2
=
,
X ,
[17]); in p a r t i c u l a r , each
a £ Z C .
respect to t h i s triangulation. where
(C/V )/F ,
Vf
J
=
F
(= V/V )
|F|
=
V )
as
V acts on
C i s then
.
+ x II .
= 0
in
R
which gives r i s e to the C
(see pp. 7-8 in
C/V C/V .
£ )
V i s then simplicial with
inherits a c e l l s t r u c t u r e , with Similar remarks obtain for V .
Since
C/V =
acts freely, i t follows that
C/V
C/Vf
lying over each c e l l
.
N (E f ) = kN.(E) , as asserted above. 3 3 2 On the other hand, since C/V! = T , e(C/Vf)
=
0
= N^E 1 ) - N2(Ef) + N 3 (E f )
.
It follows readily from Lemma 3.^- in [18] that there is a Vf
as above
such that the cell structure on C/V1 is actually a simplicial triangulation.
Since each triangle has 3 edges and each edge lies on
precisely 2 triangles, 2N 1 (E I )
=
,
Thus ,
(= set of j-simplices in
number of cells of of
Thus,
+ x
i s any group of f i n i t e index in
where k
I ,
The action of Thus
!
also,
such
a two-dimensional t o r u s .
N.(E) = number of (j - l ) - c e l l s on f
II : x
induces a triangulation on the space
gives a (j - l)-simplex on
C/V! ,
K ;
log x 2 , log x )
Note that the Ehlers 1 complex manifold
R
K (and hence
With t h i s action,
• (log x l9
n/Z 2
in
by
C i s homeomorphic to the plane
*
such that:
X/V , j > 0 ;
(x_, x^9 xO
V i s isomorphic to a l a t t i c e of rank two in C/V
in V
E' =
by the three real embeddings of
"by coordinate-wise multiplication.
V-invariant.
By
x_xoxQ = 1 •
and
k
N 3 (E f ) .
=
C denote the set of points
x. > 0
embedded in
To prove the lemma we will
of finite index
= kN^(E) , where
2N 1 (E f )
(ii)
78
3N.-(Ef ) = 2Np(E f ) . Consequently, 2N 2 (E») - 2N3(E»)
= N^E1 ) .
This proves (ii) above, and hence shows that other hand, by the above Fact,
2N.(E) = N^(E) . On the
Thomas & Vasquez: Betti numbers of varieties b3 since
=
e(E) = N
2b^ - e(E)
=
2S± - N 3
=
0
79
,
by Theorem 12 of [3]. This completes the proof of
Lemma 1, REMARK, 3
Theorems 3 and k hold even if
T
has torsion.
BETTI NUMBERS M D HODGE NUMBERS We illustrate the preceding material with two sets of
examples, both coming from cubic number fields. (totally real) field with special modular variety
X, e ,
V
Let
K
a group of modular type.
Z?
for
1 3 and c. . Since b 2 = J(e + b 3 ) - 1
be such a In [19], for any
T , we have computed the Chern numbers
,
(3.1)
bp(Zp) can be readily calculated from e and x » "by Theorem 2, Also, by Theorem 3, b (Z~) = 0 , since E has the homotopy type of a U-complex and b^Q) = 0 by (3.M of [19]. Consider the family of Galois cubic number fields K = Q(X) , where X3 + (n + 1)X2 + (n - 2)A - 1 = 0 For
n = 1, 2, 3, h9 6, 9, 10
[6] and hence group for
Y
K ) ,
or
12
,
n> 0 .
these fields have class number
has but a single cusp ( G
1
denotes the Hilbert modular
In [IT] we give an explicit resolution for this cusp,
while in [19] we give explicit resolutions for all quotient singularities that arise for cubic number fields, modular variety fbr f
=
G
Let
z(f)
denote the resulting
, where
conductor
K
=
n(n - 1) + T
In the following table we give the Chern numbers for these eight varieties, and then calculate the Betti numbers indicated above.
h
is Moishezon (it dominates
^P'^ = h 9 -^
and
and
We also calculate the Hodge numbers
The Hodge numbers Z-p
b2
b. =
h 2 '°(Z r ) = 0 ; and
£
h3>0
as and
are calculated as follows.
h » . Since
Yp , a compact protective variety),
hp>q-
h1'0 = 0
^
bo
[2], Also, Freitag [5] shows that
since
b
= 0 .
Thus,
Thomas & Vasquez: Betti numbers of varieties TABLE: Betti Numbers and Hodge Numbers of e
f
X
7 9
1
13
1
19
1
37 79 97
0 -11
-762
190
-2k
-1I+20
156
139
-82
-1*218
-234
If
X
Z(f)
(3.3) 2
2
h '
1
6k Ik
31
*3 0
0
0
36
0
0
0
-k
56
27
0
0
0
-6 -6k
Ik
36
0
0
0
100
53
1
3
1U2
8 96
12
36
177
200
25
75
2lU
66k
83
2**9
-6 -6
1
b
80
is a rational protective variety, then
Thus, for
f = 7, 9, 1 3 , or
19
the varieties
Z(f)
rational,
(it is shown in [22] that if conductor
x( x )
=
1 •
are possibly
K > 19 , then
X(G^) ^ 1 . ) Thus (as noted in [22]) we have: Problem 1. variety
Z(f)
For
rational?
By [12], for
f = 7, 9, 13,
or
19
is the Hilbert modular
If not, is it unirational? f = 37, 79, 97
and
139
the variety
Z(f)
is
of general type. Note also from the Table that the same invariants. Problem 2.
Z(9)
and
Z(l9)
have precisely
Thus, we have: Are
Z(9)
and
Z(19)
diffeomorphic?
If so, are
they biholomorphically isomorphic? In the study of Hilbert modular surfaces it has proved useful to study surfaces subgroup of
Z^
where
T
is a torsion free principal congruence
G . We now calculate the invariants given in (3.3) for three
such examples, K
cubic.
Here
T(P)
denotes the (torsion free)
principal congruence subgroup associated to a prime ideal ( = (X - r) ) , see (1.7) in [19].
P
Thomas & Vasquez: Betti numbers of varieties
81
(3.U)
TABLE: Betti Numbers and Hodge Numbers for ^-piP)
k
4
f
P
X
7
(A + 3)
-2
13 19
(X + 2)
-5
-lM
80 80
(X + 3)
-165
-7080
-360
e
b
h 3,0
b
3 2U
2
51 63 U83
3
U8
6
9 18
1328
166
U98
BIRATIONAL ISOMORPHISM If
V
and
classical notion of "birational") —
Vf
V
are complex protective varieties, one has the V1
and
being birationally isomorphic (or just
e.g., see [9]. Since we wish to consider modular
varieties that are not necessarily protective it is convenient to extend slightly this notion.
If
V
Vf
and
has introduced the important idea of equivalent [lU]. Suppose that complex manifolds.
V
are just complex spaces, Remmert V
and
and
We will then call
V
f
V
being bimeromorphically
are compact non-singular
V
Vf
and
birationally
isomorphic if they are bimeromorphically^ equivalent, considered simply as complex spaces (compare [l]9 page ^93). Let modular type.
Zy
be a Hilbert modular variety for
such that
p
is
to
Y~ - S , where
S
is
the set of singular points in
Yp . Moreover
Zp
is compact and non-
singular.
Thus, if
Z'
birationally isomorphic.
p
from
is a second model for
?„
a group of
Z-, - p" (S)
Thus there is a map
a biholomorphic isomorphism from
to
T ,
$„ ,
(See pages 13-17 of [20]).
Zp
and
Z»
x(zr) = x(zf) by Corollary 2.15 in [20]. n-manifold
X
(U.D (Following Hirzebruch [10], for any complex
we set
arithmetic genus
X = x(X)
=
n I
(-l) 1 dim H1(X; n )
i=0 where
Q,
are
In particular,
is the structure sheaf for
X
,
X ).
For use in §5 we show (compare, [7] page U9U). PROPOSITION U.2.
for
Then,
Let
Z
and
ZT
be two non-singular models
Thomas & Vasquez: Betti numbers of varieties Proof. isomorphic.
As noted above,
Z
ZT
and
82
are birationally
Thus (see [20]) there are open sets if
i
uf C vf C zf
U C v C z , and (holomorphic) maps • Z1
f : V
ff : V f
,
• Z
such that codim(Z - U)" > codim(Z» - U f ) f(U) = U f
>
1
,
1
,
codim(Z - V) codim(Zf
f!(Uf)=U
,
- V)
>
2
;
>
2
;
f|u = (f'lu1)""1 : U « U f
,
.
U = Z - p" 1 (S) , U f = Z r - p t - 1 ( S ) ) .
(In fact,
Therefore, ^ ( V , #) Set
= f # o i j j
and
(|>o())» = i d ^(U,
^(Z, (j>» = f £ o i ^ ~
9
TT ( U , * ) .
)
TT ( Z , * ) .
5
*)
TT^V,
Clearly,
^(Z',
<()' o <|) = i d
on
*)
.
TT ( U , # )
Since
^ ( Z , #)
and
f
are onto i t follows that f
, .
1
on #
*)
c() o(|) = i d
on
^(u
1
TT ( Z , # )
, #) and
•^(Z
1
,
#)
(p^ocj)' = i d
on
This completes t h e proof.
PROOF OF THEOREM 1 As noted in §k9 the fundamental group is a birational
invariant, and so we may assume that method.
Write
Z^
constructed from
as in (2.l)?and set
Z
Q.fs .
and then the
Z^
isotropy subgroup of
in
h + s
is constructed by Ehlers' Z
= Y^
so that
Z^
is
stages, adding first the successive
We take the first cusp to be °° ) C G^ ,
°° ; thus
and so by page 235 of [ 11},
IV T^
E.fs
(the is
(isomorphic to) the group of matrices 2 e
where
e
y
is a unit in
e
=
0
and
principal congruence subgroup
1 mod A
A T .
,
is the integral ideal that defines the
Thomas & Vasquez: Betti numbers of varieties
Set {E , . . . ,
E
Z = Z.
h,Q-,9
U p
K—J.
K.
...
{P. } = k
To prove the theorem i t suffices to show:
V z i 9 *^
(a)
l < k < h + s , where
K.
Q} .
9
83
= 1
M We "begin with assertion (b).
(For simplicity we now write simply TT(X)
for
P
7T (X, *) ) . Note that 1
•
K.
H Z
k—A.
= 3P
k
;
also,
its interior the angular set ( = exceptional divisor) P
k
- S
has
k
contains in
k
.
K.
3PV
as a strong deformation retract and since
•&
real codimension
P S
2 , ,IT( 3P )
• ir(P )
k
is onto.
Since S
k
has
Thus, (b) follows at
J£
once from t h e Seifert-Van Kampen Theorem ( e . g . , [ 1 3 ] ) . To prove ( a ) , s e t E = E and consider t h e diagram: ir(3E)
As above
j ^ is onto, and so by Seifert-Van Kampen, TT(Z ) ^
(d) Note that
3E = R n " 1 /r o o
TT(3E) = T^ .
Similarly,
i^ : TT(3E)
* TT(E)
i#
»• IT(E)
TT(Z)/normal closure of
(see page 191* [ll]), where
T^
.
acts freely, so
Tr(ZQ) = Tr(Yr) % T . Moreover, we may identify
with the natural inclusion
is a monomorphism.
i^(Ker j # )
1^ C r ;
in particular,
Thus, by (d) we will prove Theorem 1 when we
show: F
=
normal closure of
i^(Ker j # )
Consequently, by a result of Serre (Cor. 3, page ^-99, in [15])» Theorem 1 is proved when we prove the following result. Let that
be a resolution of any cusp
3E = R ^ ^ / r C M , V) , where
, 0
E
with
v
in
V
and
m
T(M, V)
in
is the group of matrices
M . Thus,
1
Regard
M
as a subgroup of
1 ml
(M, V) , as above, so
F(M, V) by the map
TT(3E) % T ( M , V) .
Thomas & Vasquez: Betti numbers of varieties
84
LEMMA 3. Let i denote the inclusion 8Ec — + E . Then, with the above identifications, M C Ker j # . "Proof. Recall that
0
1
5. = J K . + * x
£ R
K C R .
.
J
3
Here
T(M, V) acts on H n by the formula:
d e n o t e s t h e image of
we w r i t e
T = T(M, V) ,
x
in
K
by t h e
and d e n o t e by
1
j
p , p
imbedding
tf
and
p
the
c a n o n i c a l p r o j e c t i o n maps P!
H* V : H11
Define
Pf
^H^/M • R+
T
>lP/T
,
p
=
p"op'
.
by
For any c in R + , 3B = V~ (c) is T-stable, and for a fixed identify
9E with
9B /T 9
c we
as above.
Let b ^ 3B ; covering space theory gives an isomorphism (which depends on b )
°b : r * \^E We note for later use the special case, ®v,(m) = [P ° ^ -1 where
y
: [0, l]
• 3B
m
for
me M ,
by
c t 1 >b + tm ( = b ^ ' + t m ^ ' , in the j t h coordinate, 1 < j
V , Xy the associated complex n-manifold, and Fy the "exceptional divisor" in X y
(see §1 in [3]). There is a biholomorphic map
7T : X^ - F^ » Cn/M ;
and if we set
X = T T ^ I ^ / M ) U F^ ,
then the desired resolution for the cusp notation we also write T T ^ P ' C B )) U
Let (v , ... , v ) coordinates
-t
2J
is
(M, V) . By an abuse of
E for the manifold with boundary,
, where
F y
C
E = X/V
= {v-1( d) E ^\
B
c
< d G RJ
C
.
"•
b = TT ( p f ( b ) ) .
There i s then an n-simplex
in
b ^ (C )
E such t h a t
(z , . . . , z )
in
C^ ;
^ X.
cr =
Suppose t h a t
r e c a l l t h e formula for
b
has
TT on C^
Thomas & Vasquez: Betti numbers of v a r i e t i e s
where
Wj
=^
n
1
Iog(z 8 )v (J)
I
Writing
z s = r se
S"~_L
n j
a\
s = 1
Define
y
2TT
s=l
: [0, l ]
27Tit The above f o r m u l a e f o r 9B /F
v
9 (v, ) D
K.
in
IT (3E, p(b)) . -L
p 6,
Since
n
oiroy
< i s a loop in
shows that this loop {v , ... , v } JL n
M , to prove Lemma 3 it suffices to show that each loop
deformed to a constant loop in C z ) x C x (z ... (z f
l
. . . , zn)
show t h a t
and the formula for the isomorphism
represents for
TT and
this becomes:
-log(r
I
b
85
•••
J£-~-L
K. i"_L
9
is a basis y
can
^e
H x . This deformation takes place in z ) , as suggested by the picture: n
Figure 1 The formulae for TT and V show that t h i s contraction in fact takes place in Cn n x . (Recall that the v {j } are a l l positive numbers and that i f (z , . . . , z ) £ C H X then so does (z 9
l
•"•
z
k-l»
Z
k+1'
for a l l
w G C with
0 <
< k
Thomas & Vasquez: Betti numbers of varieties
86
This completes the proof of Lemma 3 and hence of Theorem 1. Proof of Lemma 2. free Z-module of rank discontinuously on
As noted above
n - 1
X ,
CLAIM.
X
Proof.
Let
U
v(a.) ,
Since
V
is a
Lemma 2 follows at once from:
is simply-connected*
V(a) = X n ( c n ) a
t h a t of E h l e r s , [ 3 ] , page 13*0. X. =
E = X/V .
which acts freely and properly
a E Z(n) .
for f
a s
Order t h e
11
in
Zr '
(Notation i s so t h a t i f
then
X = U X.
and
X
Since any compact subset of
X
suffices to show that each
X.
O V(a
is in
X.
.)
for
is 1-connected.
i j
0
.
large enough, it
For this the Seifert-
Van Kampen Theorem and the following Lemma suffice. LEMMA U.
(i) (ii)
V(a )
is contractible9 all
V(a ) n X.
k .
is arcwise-connected, all
The proof follows from the definition of the manifold
k . X
given in [3], page 13^-. We omit the details. REMARK.
We take this opportunity to correct several misprints o ... + (n + l)X +
in [19]. On page 19U, line 15 should begin:
(n - 2)X ... , while in line 21, [20] should be [2l]. Also, the parenthetical sentence in line 19 should read: (...for all other (with
F(n)
not
one of the above
8
fields), . . . ) .
n
Finally, on
page 205, line 11, Figure 3 should be Figure 2 Research supported by grants from the National Science Foundation. REFERENCES [1] [2] [3]
Bredon, G. (1972). Introduction to Compact Transformation Groups. New York: Academic Press. Deligne, P. (1971). Theorie de Hodge, II. Publ. Math. I.H.E.S., 40, 5-58. Ehlers, F. (1975). Eine Klasse komplexer Mannigfaltigeiten und die Auflosung einiger isolierter Singularityten. Math. Ann., 218, 127-156.
Thomas & Vasquez: Betti numbers of varieties [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
87
Freitag, E. (1972). Lokale und globale invarianten der Hilbertschen modulgruppe. Invent. Math., 17_, 106-134. Freitag, E. (1975). Singularitaten von Modulmannigfaltigkeiten und Korper Automorpher Funktionen. Proc. Int. Congress Math., 443-448. Gras, M. (1975). Methodes et algorithmes pour le calcul numerique du nombre de classes et des unites des extensions cubique cyclicques de Q. Journal f. d. r. u. a. Math. (Crelle) , 227, 89-116. Griffiths, P. & Harris, J. (1978). Principles of Algebraic Geometry. New York: John Wiley & Sons. Harder, G. On the cohomology of SL(2, 0). Ln Lie Groups and Their Representations, 139-150. Hartshorne, R. (1977). Algebraic Geometry. ^E Graduate Texts in Mathematics, no. 52. New York: Springer-Verlag. Hirzebruch, F. (1966). Topological Methods in Algebraic Geometry, 3rd ed. Berlin: Springer-Verlag. Hirzebruch, F. (1974). Hilbert modular surfaces. L'Enseignement Math., .19,, 183-281. Knoller, F.W. Beispiele dreidimensionaler Hilbertscher Modulmannigfaltigkeiten von allgemeinem Typ. To appear. Massey, W. (1977). Algebraic topology: an introduction. JEn Graduate Texts in Math., no. 56. Berlin: Springer-Verlag. Remmert, R. (1957). Holomorphe und meromorphe Abbildungen komplexer Raume. Math. Annalen, 133, 328-330. Serre, J.-P. (1970). Le probleme des groupes de congruence pour SL 2 . Ann. Math., j^2, no. 2, 489-527. Svarcman, O.V. (1974). Simple-connectivity of a factor-space of the Hilbert modular group. Functional Analysis and its Applications, _8, 99-100. Thomas, E. & Vasquez, A. (1980). On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields. Math. Ann., 247, 1-20. Thomas, E. & Vasquez, A. (1981 ) . Chern numbers of cusp resolutions in totally real cubic number fields. J. Reine Ang. Math., 324, 175-191. Thomas, E. & Vasquez, A. (1981 ) . Chern numbers of Hilbert modular varieties. J. Reine Ang. Math., 324, 192-210. Ueono, K. (1975). Classification theory of algebraic varieties. In Lecture Notes in Math., no. 439. New York: Springer-Verlag. Vigneras, M.-F. (1976). Invariants numeriques des groupes de Hilbert. Math. Ann., _224, 189-215. Weisser, D. The arithmetic genus of the Hilbert modular variety and the elliptic fixed points of the Hilbert modular group. Math. Ann. To appear.
HOMOTOPY
PAIRS
IN ECKMANN-HILTON
DUALITY
K.A. Hardie
0.
Introduction
In a long series of papers appearing over the seven years 1958-64 Eckmann and Hilton studied in depth a partly heuristic duality in the homotopy theory of pointed topological spaces. Hilton has recently written a retrospective essay
[13]
discussing their motivation and philosophy including, together with a survey of more recent developments, a rather complete list of papers having a bearing on the subject. Basic to the duality was the concept of pair.
Classically
this derived from relative topology whose objects were indeed pairs of spaces
(X, A)
with
A ex.
pair is associated an inclusion map
However with every such A •> X.
On translation
into categorical language the arrow is more important than its domain and codomain.
Thus (and for other good reasons) a pair
came to mean a (pointed) continuous map.
In formulating the
naturality properties of the two exact sequences in homotopy associated with a pair it was necessary to consider maps of pairs. a map
From relative topology the obvious concept to take for f •* g
was a (strictly) commutative diagram
A (0.1)
f X
and later when contemplating homotopy of maps of pairs it was natural to consider homotopies satisfying
<J> f = gi|>
for each
<|>, : X •> B t € I.
and
ty
: A •> E
Yet, as has been
discussed in
[8], [9] and [10] the associated pair-homotopy
category
is not from all points of view the most convenient
PC
Hardie: Homotopy pairs
89
one to study.
To mention two considerations only : the
morphism sets
[f, g]
in the pair-homotopy category are not
always invariants of the homotopy classes of
f
and of
g;
the notion of homotopy equivalence of pairs that features heavily in the work on the Puppe sequence [18] is weaker than that of pair-homotopy equivalence.
For these and other reasons
the authors of the papers cited have considered homotopy commutative diagrams. A homotopy pair map from
(f
, {h.}) where
h
f
to
g
is a triple
is a homotopy from
<j)f
to
g\p and
{h } is the associated track or homotopy class of homotopies from <j>f to gip. That this definition gives rise to a category (in fact to a Zisman [3], f
to
g
HPM
2^category) was already observed by GabrielThe set
Tr(ff g)
of homotopy pair classes from
is obtained by factoring out by the relation
where the square referred to on the right is the composite in the obvious sense of the three internal squares. resulting category
HPC
In the
homotopy equivalent pairs are isomorphic, f /vr> Pf
the Puppe construction
is functorial
(Pf
is the pair
that includes the codomain of
f
P
is the pair that maps the fibre
of
is left adjoint to f
N
(Nf
into the domain of
similar vein every pair
into the cofibre of
f) : for details see f
is isomorphic in
( the inclusion of the domain of
f
HPC
[9]. to
f)
and
In Jf
into its mapping cylinder),
thus every pair is isomorphic with a pointed cofibration.
Hardie: Homotopy pairs
Alternatively we may regard PC
90
J
as a functor from
HPC
to
which then is a full and faithful left adjoint to the
obvious functor every pair
f
I : PC + HPC.
For details see [8]. Dually
is isomorphic in
of the mapping-track of
f
HPC
to
Lf
(the projection
onto the codomain of
f ) , thus
every pair is isomorphic with a Hurewicz fibration. L : HPC -> PC
Moreover
is a full and faithful right adjoint of
I.
Since homotopic maps are evidently isomorphic as pairs the homotopy invariance of
ir(f, g)
the relationship between observing that if
f
and
7r (f, g) = Tr(f, Ig) «
[12; 3.6 ']
[Jf, g] ?
f
and
Jf
have
so that, since
we obtain the following
If
f
is an h-eofibration
h—fibration Suppose now that
C
then
C
or if
i\ (f, g) «
C.
g
is an
[f, g] .
is an arbitrary category and let
the associated category of pairs of
P
associated reflector with coreflector
P.
Moreover
f /\/-* codomain f
f 'v+ domain f.
C
P
is abelian.
as
and also coreflective C
are often carried
For example it is well-known [17] that if
is abelian then
C
is reflective with
The embedding is the more
interesting because special features of P.
be
Regarding each
as an identity pair enables us to regard
a full subcategory of
over to
is clarified by
[8; Corollary 2.6].
Proposition 0.2
object of
On the other hand
[f, g]
is an h-cofibration then
the same pair-homotopy type proposition
is assured.
7r(f, g)
C
It was clearly the intent of
Eckmann and Hilton to extend concepts and, where possible, theorems into the category of pairs. in which
C
Returning to the case
is the pointed homotopy category, Proposition
0.2 implies that the composite
C + PC I HPC
is an embedding
since identity maps of spaces are cofibrations. has been observed in
[9; Theorem 7.1]
C
(respectively coreflective) subcategory of reflector
/
f v»- codomain f
f r^-+ domain f) .
Moreover as
becomes a reflective HPC
via the
(respectively coreflector
The argument is of course slightly different
but the effect is that the program of Eckmann and Hilton still makes sense.
In this spirit we examine in §2 the concepts of
H-space and
1
H -space and the interesting phenomenon alluded to
Hardie: Homotopy pairs
in [13] that whereas
X
91
is an H-space if and only if the unit
of the loop suspension adjunction at whereas
X
X
is a
coretraction and
f
is an
H -space if and only if the counit of the
loop-suspension adjunction at
X
is a retraction, the known
proofs of these results do not dualize. relevant extensions of the concepts to
We show that the HPC
already exist but
have not been recognized and we obtain corresponding retraction characterizations in terms of the
N-P
adjunction.
We also
consider the question of the appropriate definition of Lusternik-Schnirelmann category and cocategory of a map.
We
show that the method of Ganea [4] for defining cat and cocat via constructing sequences of maps essentially exploits the N-P
adjunction. Eckmann and Hilton [2] succeeded in deriving all the then
known exact sequences of homology, cohomology, cohomotopy, etc. as special cases of two dual exact sequences of morphism sets in
C
and
PC.
In § 1 we show that these can be studied
equally well from the point of view of
HPC,
that in fact some
of the relevant proofs become somewhat simpler.
1. If
Z
Exci sion morphi sms
denotes the (reduced) suspension functor we shall ^ n ( f , g) = iT(£nf, g)
use the notation
to distinguish morphism sets in X, Y
HPC
and and
^ ( f , g) = [£ n f, g] PC.
we have, as remarked in the introduction,
IT (X, Y) .
If
g : E -> B
For spaces IT (X, Y) =
is a pair then Eckmann and Hilton
[2] define n n (X, g) = [Z n ~ 1 i, g] ,
(1.1) where X.
i : X •> CX Since
CX
recognize that
Let
X
into the cone on
is the cofibre of the identity pair i = PX n
*X
n'
X
X,
we
and we have the following proposition.
' g * w ^n-i' X ' F Q ^ ' wlaere the homotopy fibre of g.
Proposition 1.2 Proof
is the pair including
F
Q
denotes
denote the inclusion of the base point
*
92
Hardie: Homotopy pairs
into 0.2
X.
Since
we have
Ng) ^
i
is a cofibration, applying Proposition
II ( X , g ) = TT n
n~~ i
<-y
( P X , g ) « ir
1
n*" i
irn_1 (*X, N g) = IT _ (X, F ) .
( X , N g ) = TT
1
n~ i
(P(*X)f
The last equality
follows directly from the respective definitions.
Alternatively
it is a special case of [10; Lemma 1.2 (i)]. Remark 1.3
Proposition 1.2 enables us to recognize the
Eckmann-Hilton sequence
nn(x, E )
S*(g)
nn(x, B)
[12]
n n (x f g)
n n _ 1 (x,
as being essentially the dual Puppe sequence
V X ' B)
E) Of course
S^(g)
V i l x ' Fg)
Vi(x'
B)
remains convenient as it lends itself so
elegantly to specialization.
We need not however provide a
separate proof of exactness. Remark 1.4
The reader should be warned that the notation 1.1
is ambiguous.
If
X
is interpreted as the identity pair, which
is certainly a cofibration, then TTn(P(*X), g) «
7r
n<*
Proposition 1.2.
x
N
' 9>
=
3
""n* *'
II (X, g) « E)
in
ir (X, g) =
conflict with
For the remainder of this section we shall
use the interpretation 1.1. Remark 1.5
Proposition 1.2 can be used to derive the
formulae [12; 3.1'] : (1.5.1)
if
g
* •> B
then
(1.5.2)
if
g
B •> *
then
Remark 1.6
-1 F = g '(*)
Let
inclusion map.
IIn(A, g)
n n (A, B) .
, g) « n n-1 (A, B) . and let
v : F -»• E
denote the
Then the pair maps
and
induce the excision morphisms
and
n
-1
F
>
e" : II (A, v) •> IT (A# B) respectively and we obtain
Hardie: Homotopy pairs
93
the following short proof of [12; Theorem 3.3']. Proposition 1.6.1
Proof
If
g
If
g
is a fibre map then
are
isomorphisms.
v
IT (A, g)
Similarly
e"
^
s
e"
[12;
is homotopy equivalent to
^ ( A , v) « 7 ^ ^ (A, F v ) & T T ^ (A, F
7T (A, B) .
and
is a fibre map then it is well-known
Corollary (3.7)", dual] that Hence
e"
Ng.
) « TTn_1 (A, QB) »
equivalent to
TT
(A, F ) ->
which is an isomorphism by Proposition 1.2.
2.
Cyclic and cocyclic maps
In this section we shall assume that all spaces have the pointed homotopy type of a
CW complex.
We recall that a map exists an associated map
f : A -> X
X x A
(2 1)
'
is
F : X x A •> X
3 f
cyclic if there such that the diagram
•X
^ ^ X
v f)
X v A is homotopy commutative, where the codiagonal map.
j
is the inclusion and
V
Cyclic maps were first studied by Gottlieb
[6] and Varadarajan [20]. A convenient survey of the known results can be found in [15]. The property of being cyclic is invariant under isomorphism in Lemma 2.2
If (<|>, \p, t n t H if
: f -»- g
Indeed we have : is a homotopy pair map,
<|> has a left homotopy inverse
is cyclic then If
HPC.
f
F 1 is a map associated to
associated map for
y
and if
is cyclic. g,
yF'fcf) x ^)
choose
as
f.
We may clearly associate with
(<|>, ty, {h } ) : f -> g
((> : domain (f) -> domain (g) .
This rule gives rise to the
domain restriction functor
d : HPC -• HC.
Let
n : 1 •> NP
g
denote the unit of the
we have the following theorem.
NP
the map
adjunction.
Then
94
Hardie: Homotopy pairs
Theorem 2.3
A map
f : A -* X
dnPf Remark 2.4
-£s cyclic if and only if
has a left inverse in
HC.
Theorem 2.3 can be regarded as an extension to
the pair category of the result that
X
is an H-space if and
only if the unit of the loop suspension adjunction at a coretraction, for we have NP 2 X « (fi£X)*
and
dnPX
PX « X*,
X
is
P X « *(£X),
is the unit of the
QZ
adjunction.
Before proceeding to the proof of Theorem 2.3 it will be convenient to recall some details related to the functprs and N.
There is a commutative diagram
P2f
Qf
"Pf where
P
Qf
shrinks
[18; (5)].
Since
is functorial.
X P2
to
Rf *
-* EA and
Rf
Moreover,
Rf
a : P
(Nf)~1(*) = fix and if we denote by F
CX
to
*
being a homotopy equivalence,
we obtain a natural isomorphism fix into
shrinks
is functorial it follows easily that
Mf
•> Q.
Dually,
the inclusion of
we obtain a diagram
fiX (2.5)
Mf
and a natural isomorphism
T : M •> N .
Then in the diagram
95
Hardie: Homotopy pairs
-> GZA
MQf
(2.6)
"Pf
Rf
the lower and the two central squares are commutative. maps
(Pf) ' ,
i) and
e
(Pf)•(x) = ((Pf)(x), (x, -)) = (x, (x,-))
where
(z £ C f , X € C p f x )
e (a)
(a € A)
(a, -)
(x, -)
The homotopy a
(x £ X)
*(z f X) = (z, (Rf)X) =
indicates the path hfc
is the path
The
are given by equations
is given by
t «-• (x, t)
in
Cpf.
h t (a) = ((a, 1-t), afc) ,
s *-+ (a, 1-t + ts)
in
where
ZA.
Now Brayton Gray [7; Theorem 4.2] has proved that there exists a homotopy equivalence
<|> : (X, A ) ^ -* F
f
t
where
{X, A ) ^ is his relative version of the James construction. Examining his map iMPf) ' : X -• F followed by
f
4>
factors into the inclusion
a
dnPf
It is well-known that has a left inverse in
is a natural isomorphism,
have left homotopy inverses. is cyclic.
X •+ (Xf A ) ^
<(>.
Proof of Theorem 2.3 Suppose that
it can be checked that
ty
MQf
HC.
is cyclic.
Then, since
and consequently
i)j(Pf)1
Lemma 2.2 now implies that
Conversely suppose that
loss of generality we can assume that
f
is cyclic. f
f
Without
is a cofibration
(and therefore an inclusion) and that the corresponding diagram 2.1 is strictly commutative.
Then
A
acts
on
X
in
Hardie: Homotopy pairs
96
the sense of [7; Definition 3.2] and hence, by [7; Theorem 3.2] (Pf) f
has a left homotopy inverse. A space
X
admits an H-space structure if and only if its
cocategory in the sense of Ganea [4] is less than or equal to 2.
Ganea's definition can conveniently be expressed in terms
of the f
PN
adjunction as follows.
Let
is already defined, let
f
. = dnf .
equal to the least value of
n
for which
homotopy inverse, if no such f n Since
X* « PX,
£* = X*,
Then cocat X f
exists then
a natural definition for
obtained by setting Corollary 2.5
f1 = Pf
A map
and
f
f : A •*• X
and if is
has a left cocat X = °°.
cocat f
. = dnf
is
as before*
is cyclic if and only if
cocat f < 2. The notion of cyclic map can clearly be dualized as follows.
A map
g : X ->- A
is cocyclic if and only if there
exists an associated map
G : X •*• X v A
is homotopy commutative.
The property of being cocyclic is
HPC
such that the diagram
invariant for we have a dual of lemma 2.2.
Lemma 2.7
If
g
is cocyclic, if
is such that then
f
(cj), \p, {h }) : g •+ f
\p has a right homotopy inverse
is cocyclic.
A dual to Theorem 2.3 can now be stated in terms of the codomain restriction functor e : PN -> 1 Theorem 2.8
A map ceNg
Remark 2.9
c : HPC -* HC.
denote the counit of the
Since
g : X •> A
NP
is cocyclic if and only if
has a right inverse in ceNX
we recover the result that
Let
adjunction.
HC.
is the counit of the X
is an
HC.
adjunction
H -space if and only if
the counit of the loop suspension adjunction at inverse in
Ql
1
X
has a right
Hardie: Homotopy pairs
97
Proof of Theorem 2.8 Suppose that ceNg h a s a right inverse. It is well known and essentially due to Puppe [18] that Qf is cocyclic for every m a p f. Let h : C •* X N be a representative of ceNg. Then since ^ g . h . PN g * 0 and IF. P N 2 g there exists Ng is the cofibre of 1 a m a p h : EF. A such that the lower rectangle of the Ng following diagram is homotopy commutative.
Applying Lemma 2.7 w e have that g is cocyclic. Conversely suppose that g is cocyclic and consider the standard homotopy pullback diagram : X
(g x X)A
" *
A
An associated m a p G provides a homotopy diagonal for the above rectangle and therefore a homotopy section for TT exists. To complete the proof it is sufficient to show that TT is isomorphic in HPC to h. The following argument modifies and specializes the construction of Gilbert [5; Proposition 3 . 3 ] . P is the total space obtained by converting j into a fibration and pulling back over (g x X)A. Thus P is the relative path space E(A x x, A g , A v X ) , where Ag c A x x is the image of (g x x)A, and TT (?) = TT where TT. X is the
Hardie: Homotopy pairs
projection. we have Ng
In view of the natural isomorphism
h « k,
and maps
k
where
CftA
to
and
k : CM *.
into the fibre map U=
{(sy, v) € F
Then
v
U CftA
X
x X
U CftA •> X
To compare
k
and
T : M -> N , agrees with ir ,
convert
where x X 1 | k(sy) = v(1)}
U CftA c CF
V = { (sy, v) € F A : {(y, v) € F
= F
v : U •> X,
v(sy, v) = v(0).
Let
98
has fibre 1
k(sy) = v(1) , v(0) = * } .
| Ng(y) Ng (y) = v(1) v (1) }
lifting function for the fibration
Ng
F
be a path
and let
w : U
P
be given by
, v) ,
w(sy, v) = (A(y, -
P)
where for any point that
e
s F. , % £ A
is the path such
Then the right hand square in the
(t) = p(st).
following diagram commutes.
tth x E(X, *, X) U
E(A x x f *, A v X) Hence
w
induces a map between the fibres, which can be
factored into two maps
w1
and
w",
where
1
w (sy,v) = (sA(y, -v) (1), v) v) = (£ , v ) . By the arguments used in the proof of [4; Theorem 1.1], both w1
and
w"
are weak homotopy equivalences.
of the 5-lemma shows that
w
An application
is a weak homotopy equivalence.
and P have the homotopy type of CW-complexes N g it follows that h « k » v « IT in HPC, completing the proof
Since
C
of Theorem 2.8. Dual considerations and also the force of Ganea's
Hardie: Homotopy pairs
99
initiative now suggest the following definition of Lusternik Schnirelmann category of a map
g.
is already defined let
ceg
n
g
. =
is the least value of
in
HC,
otherwise set
Corollary 2.10
m
Let
g. = Ng, Set
for which
g
and if
G cat g = n
g if
has a right inverse
G cat g = °°.
A map
g
is oocyclio if and only if
G cat g < 2. in view of the above it is tempting to define This cannot be done however as established meaning
p : E = S m U e n U e m + n -> S n
p*i = a € H^(E),
generators in cohomology with cat (Sn) = 2, However
p
where y = aU $
a, |3, y [14]-
by a standard result we have
if such existed, would fail to preserve cup
It follows that
G cat p > 3. G cat g
and
one may give a "Whitehead type" characterization of
G cat g. and
are Since
cat p < 2.
To enable more effective comparison of cat g,
with the
is not a cocyclic map since any associated map
E •> E v s , products.
cat g = G cat g.
already has an
[1]. To see that the two do not coincide,
consider a sphere bundle property that
cat g
n-1
Let
K(n)
copies of
denote the product of one copy of X,
let
followed by projection on to
^A : X -+ n (n)
be the map which
yields
g
and which
followed by projection onto each factor
X
yields the identity
map.
Let
j : T -* IT(n)
A
A
denote the inclusion of the fat wedge.
Then we have the following. Theorem 2.11
G cat g ^ n through
Proof
Convert
j
^A
factors
up to homotopy.
into a fibration and let
denote its pull-back over fibration.
if and only if
j : T •> TI(n)
^A.
Convert also
IT : P -> X g
into a
Then it is claimed that these fibrations are
homotopy equivalent.
Indeed we need only complete the
modification of Gilbert's inductive argument in the proof of [5; Proposition 3.3], of which the first stage has already been given in the proof of Theorem 2.8. An obvious consequence of Theorem 2.11 is the following.
100
Hardie: Homotopy pairs
Corollary 2.12 If
If
g : X •* A
G cat g = 1,
g : X •> A
then
is homotopically
trivial it is easy to see that
hence we can have
also shows that
g
cat g < G cat g < cat X.
G cat g < cat A.
This example
cocyclic does not imply that
A
is an
H f -space.
3. Let
F i brations that are also cofibrations
F i E 5 B
oofibvation
be a fibration.
Then it is also a
in the sense of Milgram [16] if there exists a
homotopy commutative diagram
(3.1)
in which the map
k
is a homotopy equivalence.
Examples
have been given by Milgram loc.cit., Hausmann and Husemoller [11],
Schiffman [19] and the phenomenon studied in some depth
by Wojtkowiak
[21] .
Proposition 3.2
The homotopy class of is equivalent in
Proof
HC
k to
in diagram 3.1 cep.
First observe that the homotopy commutative left
hand square of 3.1 defines an automorphism of
i
in
HPC.
Hence replacing the left hand square of 3.1 by a strictly commutative square affects homotopy equivalence. HPC by
to
Np.
k
only by composing it with a
Next note that
i
is isomorphic in
Hence replacing the right hand square of 3.1
ep : PNp -> p,
replaces
k
by an equivalent class.
The
following is an immediate consequence.
Proposition 3.3
A fibration
p
is also a cofibration if
and only if the counit of the at
p
is an isomorphism.
NP
adjunction
Hardie: Homotopy pairs
101
Since any map is isomorphic in HPC to a fibration, the fibrations that are also cofibrations are completely determined by the subcategory of HPC orthogonal to e. Proposition 3.3 enables some elucidation of the dual phenomenon : a oofibration j is also a fibration if the unit of the NP adjunction at j is an isomorphism. It is well known that adjoint functors, restricted to the respective full subcategories of objects fixed by the unit and counit, yield category equivalences. It follows that every fibration that is also a cofibration can be obtained by applying the functor P to a cofibration that is also a fibration and then converting to a fibration. Conversely every cofibration that is also a fibration can be obtained by applying the functor N to a fibration that is also a cofibration and then converting to a cofibration. While these observations are somewhat trivial, at least they bring out the point that this is one of the happy instances in which the Eckmann-Hilton duality is categorical. References [ 1]
BERSTEIN, I. and GANEA, T. The category of a map and of a cohomology class. Fundamenta Math. 5 0 (1961/2), 265-279.
[ 2]
ECKMANN, B. and HILTON, P.J. Groupes d'homotopie et dualite". C.R. Aoad. Sci. (Paris) 246 (1958), 2444, 2555, 2991.
[ 3]
GABRIEL, P. and ZISMAN, M. Calculus of fractions and homotopy theory. Ergebnisse der Math, und ihre Grenzgebiete 35 (1967). Springer Verlag.
[ 4]
GANEA, T. A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39 (1965), 295-322.
[5]
GILBERT, W.J. conilpotency.
[ 6]
GOTTLIEB, D.H. A certain subgroup of the fundamental group. Amer. J. Math. 87 (1965), 840-856
[ 7]
GRAY, B. On the homotopy groups of mapping cones. Proo. London Math. Soc. (3) 26 (1973), 497-520.
Some examples for weak category and Illinois. J. M. 12 (1968), 421-432.
Hardie: Homotopy pairs
102
[ 8]
HARDIE, K.A. On the category of homotopy pairs. Topology and its applications 14 (1982), 59-69.
[ 9]
HARDIE, K.A. and JANSEN, operators in the category Conf. Categorical Aspects Carleton (1980). Lecture Verlag, 112-126.
[10]
HARDIE, K.A. and JANSEN, A.V. Toda brackets and the category of homotopy pairs. Proc. Symp. Categorical Algebra and Topology, Cape Town (1981). Quaestiones Math. 6 (1983), 107-128.
[11]
HAUSMANN, J.C. and HUSEMOLLER, D. Acyclic maps. Enseignment Math. (2) 25 (1979), 53-75.
[12]
HILTON, P. Homotopy theory and duality. Breach Science Publishers, Inc. (1965).
[13]
HILTON, P. Duality in homotopy theory : a retrospective essay. J. Pure and Applied Algebra 19 (1980), 159-169.
[14]
JAMES, I.M. Note on cup-products. Soc. 8 (1957), 374-383.
[15]
LIM, K.L. 349-357.
[16]
MILGRAM, R.J. Surgery with coefficients. Math. (1974), 194-248.
[17]
PRESSMAN, I.S. Functors whose domain is a cateaory of morphisms. Acta Math. 118 (1967), 223-249.
[18]
PUPPE, D. Homotopiemengen und ihre induzierten Abbildungen I. Math. Zeitschr. 69 (1958), 299-344.
[19]
SCHIFFMAN, S.J. A mod p Whitehead theorem. Proc. American Math. Soc. 82 (1981), 139-144.
[20]
VARADARAJAN, K. Generalized Gottlieb groups. J. Indian Math. Soc. 33 (1969), 141-164.
[21]
WOJTKOWIAK, Z. On fibrations which are also cofibrations. Quart. J. Math. Oxford (2) 30 (1979), 505-512.
A.V. The Puppe and Nomura of homotopy pairs. Proc. of Topology and Analysis, Notes in Math. 915 Springer
On cyclic maps.
Gordon and
Proc. Amer. Math.
J. Aust. M. Soc. 32 (1982), Ann. of
Grants to the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research are acknowledged.
The Department of Mathematics University of Cape Town Rondebosch 7700 Republic of South Africa.
PROFINITE CHERN CLASSES FOR GROUP REPRESENTATIONS BENO ECKMANN
and
GUIDO MISLIN
Eidgenossische Technische Hochschule Zurich To oixti iKlznd VztzK Hilton, on kii> blxtizth
birthday
1. INTRODUCTION 1.1. Let
G
be a (discrete) group and
a complex representation of associates Chern classes G , as follows: The map spaces induces a
G
of degree
c.(Q) € H
MacLane complex for
m . With
one
of classifying
BG = K(G,1)
G ) , and the
Q
(G;Z) in the cohomology of
B Q : BG -* BGL (c)
C -bundle over
Q : G -* GL (cl
(Eilenberg-
c.(Q) are the Chern classes
of that vector bundle, j =1,2,3,... . The bundle being flat there are many groups for which the Chern classes have finite order; indeed it is well-known that the image of H
(G;Q)
under the inclusion coefficient map
C.(Q) in
Z -* Q
which implies, under suitable finiteness conditions on that
is
0 ,
G ,
C.(Q) is a torsion element. This is trivially so if
G
is a finite group. In
that case it was shown in [5] that if the representation Q is realizable over a number field precise bound for the order of only: An integer that
K c C , then there exists a
c.(Q) depending on
K
and
j
E (j) is described in [5] with the property K EK (j)c.(Q) = O for all finite groups G and all K-repre3
Eckmann & Mislin: Profinite Chern c l a s s e s
— sentations
Q of G ; and that
104
1 —
EK(j) , or — E K ^
ding on properties of the number field
depen-
K , is best possible
in that sense. Namely, let ©^(n) for a positive integer n denote the exponent of Gal(K(£ )/K) , the Galois group over K
of the n-th cyclotomic extension
primitive n-th root of unity; then E__(j) = max{n with
K(£ ) where e__(n)
£
dividing
is a j }
In [5] the prime factorization of 1__(j) has been expressed in terms of numerical invariants attached to K . In particular, for K = Q and j even one has
E (j) = denominator
B
i of TTT where B
B. is the j-th Bernoulli number
4 ~ '20 ' B 6 ~ 42*
E (j)
etc
1 (B = — ,
-) Some number-theoretic properties of
have been investigated in Section 6 of [5]. We note
that the numbers Nogues [3]
E f j ) are equal to the
w.(K)
in Cassou-
(see also [7]),
1.2. The purpose of the present paper is to discuss corresponding bounds for arbitrary, finite or infinite, groups and their finite-dimensional complex representations. Here a result of the same generality for the order of the c.(Q) € H
(G;Z) is not known. However, it will turn out that
corresponding precise bounds can be established for the order of the profinite Chern classes denotes the ring of p r o f i n i t e Chern c l a s s coefficient number f i e l d
c. (Q)
c. (Q) € H J (G;Z) . As usual z integers,
i s the image of
homomorphism
z = lim z/nz
C.(Q)
. The
under the obvious
z -• z . One again considers a
K and a r e p r e s e n t a t i o n
Q : G -• GL (c)
. To
Eckmann & Mislin: Profinite Chern classes
formulate the result we do not assume over
K
Q
105
to be realizable
but only to have its character values in
K ; this
is natural in view of the fact (see Section 2) that the representation ring
R(G)
is isomorphic to the character ring
of
G , and that the Chern classes only depend on the element
of
R(G)
corresponding to
Q . Under that character
assumption one obtains precise bounds for the order of the C . ( Q ) , and they are again the integers 3 Main Theorem. Let A) For any group
G
K c C be a number field.
and any representation
of arbitrary degree K
E K (j) above:
m
o : G -+ GL (c)
whose character takes its values in
one has E__(j)c. (Q) = O , j = 1,2,3,... .
B) The bound
E^Cj)
is the best possible fulfilling A) .
1^3. Remarks. 1) The problem of the best possible bound for the
c.(Q) of representations realizable over
is left open. Clearly that bound is
E (j) or perhaps, de-
pending on the field, -z E (j) . Cases of number fields where it must be
E v (j)
K
K
can be deduced from the explicit
discussion in [ 5 ] , 2) The kernel of the coefficient homomorphism H -1 (G;Z) -• H 2 ^(G;Z)
consists of all elements of
H 2 ^ (G;z)
which are inf initely <^J"v:''s^^)^le * T ^ u s if there are no such elements in H
(G;z) , the Chern classes
c. (Q)
and c.(Q)
have the same order so that we get information about the C.(Q)
themselves. This is the case, for example, if the
Eckmann & Mislin: Profinite Chern c l a s s e s group
G i s v i r t u a l l y of type
(FP)
; but, of course, that
condition is too strong. In p a r t i c u l a r , cf. cerning
106
the appendix con-
C±(Q) . 3) The arithmetic group
GL (O__) , where n
ring of integers of the number field
J\
0
i s the
i\
K , is virtually of
type (FP) (see [8]). Thus for any representation Q of GL (O-J the order of c. (Q) is finite and divides E_.(j) . n
is.
~2
-K.
In particular, if t denotes the canonical representation GL (0 ) -• GL (c) for large n , the order of the Chern class n K n c.(t) is equal to EK (j) unless j is even and K formally 3 real, where it is either
E (j)
or — E__(j)
(cf. [5], Sec-
tion 5; the precise general answer is not known). 1.4. The method for obtaining bounds for the order of Chern classes of group representations is based on "Galois invariance" . One applies an automorphism representation
a of c to the
Q ; if the character takes its values in K
and if a fixes the elements of K , i.e., a € Gal(c/K) Q
and its image under
R(G)
then
a belong to the same element of
and thus have the same Chern classes. On the other hand,
the effect of a on the (profinite) Chern classes can be described explicitely. This method goes back to Grothendieck [6] where it was applied to "algebraic p-adic Chern classes" and then carried over to ordinary Chern classes by means of a comparison theorem for torsion elements. We proceed differently. In the case of finite groups a direct approach has been used in [5]. It consists in identi-
Eckmann & Mislin: Profinite Chern classes
107
fying the Galois action with certain Adams operations on the representation ring and on the corresponding flat bundles; then one has well-known properties of Chern classes under these operations. In the general case of arbitrary groups and profinite Chern classes, the effect of a € Gal(c/K) on C.(Q)
is multiplication by a unit
the action of
k (cr)
€z
determined by
cr on the roots of unity. This is shown by
applying a construction of Sullivan's [10] to representations G
of groups
of geometrically finite type (i.e., admitting
a finite Eilenberg-MacLane complex
K(G,1)). Then, we show
that for our purpose an arbitrary group can be approximated by groups of geometrically finite type. 2.. THE CHARACTER RING 2.1. A representation G
-action on
in the following a l l dimension over the group by a l l
Q : G -+ GL (
CG-modules are meant to be of
C . The complex representation ring
R(G)
of
G i s defined to be the ring additively generated
CG-modules, with r e l a t i o n s
every short exact sequence dules where
[V]
denotes the image of R(G)
V®W
CG-modules
C of Given
[V] = [V f ] + [V11]
O -> Vf -• V-*•V" -* O of
m u l t i p l i c a t i o n in over
finite
V = V(Q) o
V in
R (G)
for
cG-mo. The
i s defined by the tensor product V and
W.
one considers a composition s e r i e s 1 2
n-1
n
Eckmann & Mislin: Profinite Chern c l a s s e s Then
n 2
[V] =
[V / V
.]
where a l l
V /V
108 „
a r e simple
CG-modules, i . e . , isomorphic t o representation
Q
; and
v
point of view of
R(G)
V(Q ) f o r an i r r e d u c i b l e n [ V ] = [ © V ( Q ) ] . T h u s from t h e v v=l
all
CG-modules are semi-simple; in
other words, we can r e s t r i c t attention to completely reducible representations. By the Jordan-Holder theorem the Q are uniquely determined up to equivalence and order. The e l e tva]
ments
where the
V
are a l l inequivalent simple
modules form an additive basis of 2.2.
The character
Q : G -* GL (c) X
for
quence of passage R(G)
V = V(Q) ; we also write
%
=
xvt
+
i s a short exact se-
XVH
• Therefore the
defines an additive homomorphism
% of
into the ring of complex-valued functions on =x
image
of the representation
0 -* Vf -»• V -* V" -+• 0
CG-modules then V»—*XV
R(G)
^
only depends on
% .If
v**W
x(R(G))
cG-
G . Since
t-^is is actually a ring homomorphism; its is denoted by R (G)
and called the character
ring of G . Theorem 1. The map
% : R (G) -* R (G)
is a ring iso-
morphism. Proof. If two completely reducible representations Q. ,Q~Z G -* GL (c) CG-modules
V
(Q^)
an<
have the same character ^
v
(Qo^
a r e
x =x l e2
Q
then the
^- somor P^ i: *- c ( t h i s i s a c o n -
sequence of t h e d o u b l e c e n t r a l i z e r t h e o r e m , c f . [ 2 ] c h a p . V I I I , §12, P r o p . 3) . An a r b i t r a r y element x e R(G) can b e w r i t t e n a s x = 2[V ] - S[W ] where a l l V and W a r e simple v v v \x
Eckmann & Mislin: Profinite Chern c l a s s e s %(x) = x ( [ © v 1) ~ x([© W 1) • I f we
CG-modules. Then assume
109
x(x) = O then
®V
and
$W
have t h e same charac-
t e r and hence a r e isomorphic by t h e above remark. I t follows that
x = O€R(G)
and thus
% is injective;
i t i s surjec-
t i v e by d e f i n i t i o n .
2.3. We now consider the action of an automorphism a
of c on the representations and the corresponding ele-
ments of R(G) . The automorphism
a induces a group automor-
phism
a of GL (c) . Given a representation o : G -* GL (c) m m we write Q for the composite representation <JQ . If we c
assume that
a
leaves fixed all values of the character v
Q and QG have the same character. By Theorem 1 this
then
implies that
[V(Q)]
= [V(Qa)] € R(G) :
Corollary 2. Let Q : G -* GL (c) and
Q
be a representation
a an automorphism of c which fixes all values of
Then
[V(Q) ] = [V(Qa) ] .
Remark. In the case of a finite group known that
G it is well-
Q and Qa in Corollary 2 are actually equivalent.
This need, however, not be the case for infinite groups. For example, if Q : Z - GL4(c)
is given by
then complex conjugation
fixes the character values, but
Q
and
a
QG are not equivalent.
Eckmann & Mislin: Profinite Chern classes
2.4. The Chern classes of c. (Q) € H * (G;Z) c. (V) Let
depend on
for any CG-module c(V) = 1 + c
class. If
Q
110
(cf. Section 1.1)
V ( Q ) only; we may also write
V
(of finite dimension over C ) .
(V) + c 2 (V) +... € H*(G;z)
O -* V f -* V -• V" -• O
c (V) = c(V f ) -c(V")
be the total Chern
is a short exact sequence then
in the cohomology ring
H*(G;z)
; this is
due to the fact that every short exact sequence of vector bundles over a CW-complex splits as a direct sum. Thus the total Chern class defines a homomorphism of the additive group of
R(G)
ring
into the multiplicative group of units of the graded
H*(G;z)
. If two representations
Q-t'Qo
°^
same character then the corresponding elements of cide, and hence the Chern classes
C.(Q )
and
G
have the
R(G)
C.(Q)
coinare
the same, j = 1,2,3,... : Theorem 3. If two representations of the group
G
have the same character then their Chern classes coincide. In particular, if an automorphism values of the representation
a Q
c^f then
c
fixes the character
Q°
and
Q
have the
same Chern classes.
2- GALOIS ACTION AND ROOTS OF UNITY 3.1. Let unity in
\x (c)
denote the group of all roots of
C . For a number field
K cC
the following inte-
gers have been considered by Soule [9]: w. (K) = card{z€ \i (c) where as usual C
Gal(C/K)
with
cr* (z) = z
for all o € Gal(c/K)}
denotes the group of automorphisms of
leaving fixed all elements of
K . We write
^ c p (C ) n
Eckmann & Mislin: Profinite Chern c l a s s e s for the group of n-th roots of unity in primitive n-th root of unity, Gal(C/K)
acts on
ju
111
c ; if
5
is a
\x c K(£ ) . The Galois group
by an operation
Gal(C/K) -+ Aut \x ,
which factors through the surjective r e s t r i c t i o n map Gal(C/K) -* Gal(K(£ )/K) whose action on faithful.
or1 (z) - z
Therefore
a € Gal(c/K)
and a l l
cyclic, i . e . ,
equal to
i s , of course,
z e \x
and a l l
i s a multiple of
e (n) ,
.
\x (C)
a € Gal(C/K)
number of elements of a € Gal(c/K)
j
Gal(K(£ )/K)
The elements of j
for a l l
if and only if
the exponent of
p
fixed under
a
, for a given
, form a f i n i t e group which must be \x
^i(c)
for some
n . I t follows that the
fixed under
i s the greatest integer
n
a
for
all
such that
e__(n) is.
divides
j ; i.e., it is the integer
E__(j)
defined in the
introduction: Proposition 4. Let ments of \x (c)
fixed under
a cyclic group of order
K
be a number field. The ele-
a
for all
a € Gal(c/K)
form
E«.(j) = w. (K) , j = 1 , 2 , 3 , . . . . * D
3.2. We now consider the profinite integers z = lim z/nz ; here and throughout this section lim refers n n to an inverse system indexed by the natural numbers with their divisibility relation. An automorphism The action of a
a
of c
z >a s follows.
on \x is by the k-power map for a certain
k = k (a) t (z/nz) * . The sequence fines a unit
acts oji
k(a) € z* in the ring
k (a) , n = 1,2,3,... dez . If we put <j^: z -• Z
Eckmann & Mislin: Profinite Chern c l a s s e s to be m u l t i p l i c a t i o n by additive group of
k (a)
2/nZ
given by multiplication with
k (a) 3 = k(aD)
Note t h a t one has
Any inverse limit
, j =1,2,...
A = lim A
of
a
means of the unit
of
c
on
k(a)
k (a).
.
Z/nZ-modules i s
z-module in the obvious way. Thus an action
automorphism
of
, t h i s i s an automorphism of the
z ; i t i s the same as the inverse limit of
the automorphism of
a
112
o^
of an
A can be defined as above by
in
z .
Proposition 5. Let A = lim A be aninverse limit n Z/nZ-modules, and K c C a numberfield. Ifj for a fixed
j = 1 , 2 , 3 , . . . , an element all a € Gal(c/K)
then a
a€ A
fulfills
*>
-i
k (a) J a =* a for
is of finite order dividing
E (j) .
Proof. Put a = lim a , a € A . The assumption —-— <- n n n * n . k(cr) a = a means t h a t k (cr)-^a = a for each n and a l l n n n a € Gal(C/K) . If we choose an isomorphism of the cyclic subgroup of A
generated by a
corresponds to an element
onto a subgroup of ju (c) , a
z € \x(C) fulfilling
a z = z for
all o € Gal(C/K) . By Prop. 4 the order of z , and hence of a , divides E__(j) and so does the order of a = lim a n K *~ n n 3.3. The case we are especially interested in is that of cohomology groups
H (X;z)
of a CW-complex
X with
Z-coefficients. They are z-modules, the operation being induced by the multiplication of the coefficients; admit an action of any automorphism plying the coefficients
z
with
a
k(cr)
of .
thus they
c > through multi-
Eckmann & Mislin: Profinite Chern c l a s s e s
113
I t is known that the canonical map HNX;^)
- lim H1(X;Z/nz) n
defined by the maps
Z -* z/nz for a l l n is an isomorphism i ^ (Sullivan [10]) . Thus H (X;z) is an inverse limit lim of n * i
z/nz-modules; the action of k € z* on H (X;z) fore the inverse limit of actions on these
is there-
z/nz-modules
exactly as in 3.2 above. 3.4. We further recall the following result of Sullivan [10] concerning the action of an automorphism C
a of
on the profinite completion of BGL(c) , the classifying
space of the infinite complex linear group morphism
GL(c) . The auto-
o acts on the etale homotopy type of any complex
variety defined over
Q , in particular of the complex Grass-
mann manifolds. This action may be used to define an induced action on the profinite completion of the classical homotopy type of the variety. By a limit argument one obtains an action
cr : BGLCc)^ •• BGL(c)"
where
* denotes profinite
completion. Sullivan's result concerns its effect on (evendimensional) cohomology with
z-coefficients.
Proposition 6. The action of the automorphism C
on BGL(c)"
induces in H ^ (BGL(C) * ;z)
the coefficient
homomorphism given by multiplication of z with j = 1,2,3,... .
a of
k(cr) ,
Eckmann & Mislin: Profinite Chern classes
114
±. GEOMETRICALLY FINITE GROUPS 4.1. A group
G
is said to be of geometrically
finite type if it admits a finite Eilenberg-MacLane complex K(G,1) ; this is the case if and only if
G
(i.e., admits a finite free resolution of
is of type
z
over
(FF)
zG ) and
finitely presentable. Let sentation, o
G
be such a group, and
Q : G -• GL (c) a repre-
an automorphism of C . We consider the maps of
classifying spaces
B Q : K(G,1) -• BGL (c)
and
U
BQ
: K(G,1) -+ BGL (c) , and their profinite completions m (BQ) and (BQ ) ; moreover, the canonical map
BGL (
o
of
BGL(c)* , see
Section 3.4. Using techniques of etale homotopy (similar to those used in [4]; we will come back to this argument in a separate paper), and the fact that
K(G,1)
is compact, it
follows that the two maps K(G,1)"
^
> BGLm(c)^BGL(cr
— * BGL (
and
are homotopic; thus they induce the same homomorphism in cohomology with
Z-coefficients. The second map yields the pro-
finite Chern classes
cr
c. (Q ) € H
yields the Chern classes
2i
*•
(G;Z) , while the first map
C.(Q) multiplied by
k(a)
,
according to Proposition 6: Theorem 7. Let finite type,
G
Q : G -* GL (c)
be a group of geometrically a representation, and
a
an
Eckmann & Mislin: Profinite Chern classes
115
automorphism of c . Then the profinite Chern classes fulfill
S.(C) € H ( G ; S ) . 4.2. In order to carry over the statement of Theorem 7 to arbitrary groups we use the following approximation method. Let
G be%a group, and X = K(G,1)
MacLane complex for G . We may consider U X
of connected finite subcomplexes
an Eilenberg-
X as a union X
. By the construc-
tion of Baumslag-Dyer-Heller [1] there is, for each a, a geometrically finite group g
: K(G ,1) -* X
G
; the g
g* : H1(Xa;Z) - H ^ G ^ z )
and an acyclic map
induce isomorphisms
for all
i . Let f^ : G^ - G be
the homomorphism of fundamental groups induced by i g : K(G ,1) -• X -* X where or3 a or a The cohomology maps R1 (X;z) -+ lim H1 (X ;Z)
i
a
t*
i s t h e i n c l u s i o n map r X -»X. a
define a homomorphism
which i s an isomorphism
"
(Sullivan
i
[10]; one uses the fact that
H (X ;z)
can be given a com-
pact topology) . It follows that the canonical map - lim H 1 (X ;z) ~* nH1 (X ;z) is injective. In other a a words, the family of homomorphisms f a : G a -• G defines an i ^ i ^ injective homomorphism {f*} : H (G;z) "* nH (G ;z) , for all i : Theorem 8. Let G be an arbitrary group. There exists HNX;^)
^a family of homomorphism groups into
f : G -* G of geometrically finite
G such that
{f*} : HX(G;Z)
Eckmann & Mislin: Profinite Chern c l a s s e s is injective for a l l 4.3. group
116
i € z.
Given a representation
Q: G -* GL (C)
G we consider the representations
o = of a a
of the of the G a
in Theorem 8. Since Theorem 7 applies to their profinite Chern classes
c.(Q )
automorphism
it immediately follows that one has, for any
a of c , c.(Qa) = k(a) C.(Q)
.
Theorem 7'. The statement of Theorem 7 applies to arbitrary groups
G .
Remark. In the approximation procedure above a group G
is approximated by torsion-free groups
G
. This may
appear unnecessarily complicated, e.g., for finite groups. It should, however, be noted that Theorem 7 could by formulated for groups which are virtually of geometrically finite type, thus avoiding any approximation for finite groups. J5. PROOF OF THE MAIN THEOREM 5.1. Let field, and values in
G be an arbitrary group, K a number
Q : G -• GL (c) K . Then any
a representation with all character
a € Gal(c/K)
values fixed, and by Theorem 3 we have j = 1,2,3,... ; the same holds for
leaves the character c.(Q U ) = c.(Q) for
C.(Q) € H ^ (G;z) • From
Theorem 7 f it follows that Cj(Q)
= k(a)jCj(Q)
for all a € Gal(C/K) . This implies, by Proposition 5, that the order of
C.(Q) is finite and divides 3 proves part A) of the Main Theorem.
E__(j) , which K
Eckmann & Mislin: Profinite Chern classes
117
5.2 . It remains to prove part B) . To do this we use the calculations performed in [5]. We recall (Theorem 4.12 of [5]) that
E (j)
is the best possible bound for the order of
the Chern classes
C.(Q) of representations of finite groups
defined over the number field that it is - E (j)
if
j
K ; with the only exception
is even and
K
formally real.
Thus B) will follow if we prove Theorem 9. Let ^
j
K
be a formally real number field ,
even > 0 . Then there exists a finite 2-group
a representation
Q
o_f Q
Q
with character values in
K
and such
that \ EK(j)c.(Q) / 0 . Proof. We put
j = 2 t
with the case
6^2
6
Y + 1
€ K
(^)
>b u t
. Let £
notation of [5]. Let of order of
Q
2
with
, and
odd, 6 Jl , and first deal
y b e the integer for which * K^1)
Y+2
Q
t
.i.e.,
Y = YK(2)
be the generalized quaternion group %
a faithful
(of degree 2) . The group
c-irreducible character
Gal(K(£
.« + E
2Y+O
is cyclic of order
2
of degree
rator of ture of
2
H*(Q;z)
mal possible order
z
r
c^iax) is
a
gene-
. From the cohomology ring struc-
it follows that 2Y
=
K , of a representation of
. It is well-known that
H (Q;z) = z/2 Y
2Y
_LJ/K) +6
, cf. [5]. Thus
i|/ = I ax is the character, with values in Q
in the
c.(tip) = c b(^j)t
has maxi-
; this is precisely the 2-primary
Eckmann & Mislin: Profinite Chern c l a s s e s p a r t of
EH)
, cf.
Prop. 3.4
In the case
(d) of
6 = 1 , i.e.,
[5]. j = 2t
use t h e above c o n s t r u c t i o n of t h e c h a r a c t e r 6 = 2 .
One v e r i f i e s
that
the 2-primary p a r t of
c. (t^)
E (j)
118
with 4;
of
2Y
has order
t
odd, we Q
for
which
(the argument i s s i m i l a r
is
to
K that in the proof of Prop. 4.11 (b) of [5]) .
_6. APPENDIX: REMARK ON THE FIRST CHERN CLASS The universal bound nary Chern class representation number field
is valid for the ordi-
c (Q) itself, for any group Q : G -* GL (C)
G
and any
with character values in the
K . To show this we proceed as follows.
We first note that det Q
E (1) K
c. (Q) = c.(det Q) where
is the representation of degree 1 given by the deter-
minant of Q(g) , g eG . If all character values of Q are in K
then so are the values of det Q . Thus
through the inclusion
det Q
factors
1 : K* -• GL (c) = C* .
The multiplicative group
K* is the direct product
of a free Abelian group and a cyclic group
c
It easily follows that
E (1) . This K
implies that ding
E__(l) .
c (1) is of order 1
of order
c. (det Q ) = c (Q) is of finite order divi-
E (1). K
Eckmann & Mislin: Profinite Chern classes
119
REFERENCES [1]
G. Baumslag, E. Dyer and A. Heller: The topology of discrete groups. J. Pure Appl. Algebra 16 (1980) ,1-47.
[2]
N. Bourbaki, Algebre; Hermann, Paris, 1958.
[3]
P. Cassou-Nogues: Valeurs aux entiers negatifs des fonctions zeta et fonctions z£ta p-adiques. Inventiones math. 51 (1979) , 29-59.
[4]
P. Deligne and D. Sullivan: Fibres vectoriels complexes a groupe structural discret. C.R. Acad. Sc. Paris, t. 281, Serie A, (1975), 1081-1083.
[5]
B. Eckmann and G. Mislin: Chern classes of group representations over a number field. Compositio Mathematica 44 (1981), 41-65.
[6]
A. Grothendieck: Classes de Chern et representations lineaires des groupes discrets. Dans: Dix exposes sur la cohomologie des schemas, Amsterdam, North-Holland 1968.
[7]
G. Mislin: Classes caracteristiques pour les representations des groupes discrets; Seminaire Dubreil-Malliavin Paris 1981, Lecture Notes in Math., Springer-Verlag, Vol. 924.
[8]
J.-P. Serre: Cohomologie des groupes discrets. Annals of Math. Studies 70 (1971) , 77-169.
[9]
C. Soule: Classes de torsion dans la cohomologie des groupes arithmetiques. C.R. Acad. Sc. Paris, t. 284, Serie A (1977), 1009-1011.
[10]
D. Sullivan: Genetics of homotopy theory and the Adams conjecture. Annals of Math. 100 (1974),1-79.
AUTOMORPHISMS OF SURFACES AND CLASS NUMBERS: AN ILLUSTRATION OF THE G-INDEX THEOREM John Ewing Indiana University, Bloomington, Indiana
47405, USA
Introduction This brief essay is an illustration of one simple application of the Atiyah-Bott-Segal-Singer G-Index Theorem.
It is not intended for
experts — if you know all about the Index Theorem, stop reading now — but it JLS intended for Algebraic topologists.
The purpose is to show in a
non-technical setting how one applies the G~Index Theorem, why it is so effective, and how it can lead to surprising connections with Number Theory. The specific problem we ask is quite elementary, involving automorphisms of Riemann surfaces. which is important:
It is the pattern of application
the G-Index Theorem allows us to compute algebraic
invariants which are otherwise exceedingly difficult to calculate; the algebraic questions one then asks about the invariants are often already answered by number theorists.
This is a pattern which repeats itself in
much more complicated settings. We begin with a digression on a subject in Number Theory which has frequently arisen in Topology in recent years — the ideal class number. The Class Number The ideal class number is an invention of Kummer; its purpose was to fix-up a faulty proof of the Fermat conjecture: no non-trivial solutions for any odd prime
has
p .
The faulty proof goes like this.
We first factor
Of course, to do this we must work over the ring p-th root of unity.
x p + yP = z p
7L[A] where
We have -1 v n y) = z .
x P + yP . A
is a
Ewing: Automorphisms of surfaces
121
Next we can show (with a little effort) that the prime.
x + A y
are relatively
Factoring both sides into primes, we conclude that
p-th power in
S[A] .
x + A y
is a
Finally, one derives a contradiction; it's not
important what contradiction—it's similar to the fact that in the ordinary integers there are no consecutive p-th powers. Of course, this is wrong:
the ring
7Z[\] does not have
unique factorization in general and so one cannot conclude that the x + A y
are p-th powers.
Kummer's idea was to correct the flaw by add-
ing ideal numbers—we'll use the more modern terminology of ideals due to Dedekind. An ideal is a subset of
7L[A] which is closed under addition
and closed under multiplication by elements of take any
2Z[A] .
a e 7L[X] and consider all multiples of
cipal ideal denoted
.
Obvious example:
a . That's a prin-
The problem is that not all ideals are prin-
cipal. Now Kummer showed that ideals d_o_ factor uniquely.
The pre-
vious argument can be repeated, using ideals rather than numbers.
We
factor: <x +y><x +Ay>«•-<x + A P ~ y> = P . The ideals are still relatively prime.
We can factor both sides into the
product of primes (ideals!), and we can correctly conclude that each <x + A1y> number.
is a p-th power.
Alas, each is a p-th power of an ideal, not a
Our previous contradiction has vanished—or has it? To arrive at the same contradiction as before we need a condi-
tion which will ensure that whenever a principal ideal is the p-th power of some ideal it must be the p-th power of a principal ideal. Kummer's first great idea.
This is
Look at the collection of all ideals modulo
the principal ideals—that's a group called the ideal class group. has finite order (a theorem) and its order number. order
h
It
is called the ideal class
Our problem is solved if no element in the ideal class group has p ; that is, if
p Jf h . Then each
<x 4-A y>
is the p-th power
of an ideal, necessarily principal, and we arrive at the same contradiction as before.
We say
p
is regular if
is thus proved for all regular primes.
p ^ h ; the Fermat conjecture
Ewing: Automorphisms of surfaces
122
Clearly this would be an exercise in semantics if there were no way to determine when a prime was regular. great idea:
This was Kummer's second
he provided a simple and effective way to do this.
The class number is extremely difficult to compute and is presently only known for integers,
h = h~~h
principle) to compute. h~
p < 67 .
, and
h
But it always factors into two
, called the first factor, is easy (in
Indeed,
is the determinant of a certain
Warning:
While
h
(p + l)/2 x (p+l)/2
integer matrix
can be computed in principle, it grows very rapidly.
For p < 23 , h~ = 1 ; for ,- „ i n 33 n ~ 10
p = 23 , h~ = 3 ; but for
p = 199 ,
For his special purposes Kummer devised a special criteria for regularity, thus circumventing the difficulty of computing showed
p
(p-l)/2
is regular iff
p
Bernoulli numbers.)
h .
(He
divides the numerator of one of the first Nonetheless, the computation of
h
and
h~
has been a central theme in algebraic number theory for over a century. Automorphisms of Surfaces Now we can return to Topology...well, almost.
We can work our
way back to the present by starting with Riemann, one of the fathers of Topology. It is remarkable that in the same year (1851) that Kummer began publishing his important work, Riemann delivered his famous address on Riemann surfaces.
Of course, a Riemann surface is an analytic object,
but (surprisingly) many of its properties depend only on its topology. Towards the end of that address Riemann makes a profound observation: While Riemann surfaces are constructed in order to study multivalued maps on the complex plane, the interesting objects of study are the automorphisms (holomorphic homeomorphisms) of a given surface. It was a fruitful remark, one that has led to many beautiful and surprising results.
For example, if a surface
then the automorphism group of
S
is finite!
S
has genus
> 1
That means that for most
surfaces automorphisms have finite period. What kinds of automorphisms are there of a fixed period (To make life simple, we assume
p
is prime.)
p ?
That's a vague question
Ewing: Automorphisms of surfaces which we need to make more precise.
123
We can make it more precise by
assigning an algebraic invariant to each automorphisms — that's certainly in the spirit of Algebraic Topology. Now one studies a Riemann surface
S by studying the holo-
morphic and meromorphic functions and differentials on S . We can define our invariant by considering the holomorphic differentials. Of course, we can add two differentials and multiply by complex scalars; that is, the set of holomorphic differentials on S V . The dimension of V
forms a vector space
is precisely the genus of S ; it doesn't
depend on the analytic structure at all. An automorphism on
V .
T
of period
(If we think of a differential
bundle then
T*(u)) = w o T~
p
induces a linear mapping T*
co as a section of the cotangent
.) Of course,
T* also has order
so defines a representation of the cyclic group of order T*
p and
p . To study
we can study the character of this representation; that is, we con-
sider the trace
Tr(T*) . If T* is a linear mapping of period
the eigenvalues of T* are powers of a p-th root of unity
p
then
A . Hence,
Tr(T*) e 2Z[A] . We can at last ask our realization question more precisely. Question:
For a fixed prime
be realized as Tr(T*)
p , which* elements of 7L[A] can
for some automorphism
T
of period
p
on some
S ? That is, which characters can be realized?
Riemann surface
There is one easy restriction, known even to Riemann.
It is a
basic result of complex analysis (the "Hodge decomposition") that V e V = HX(S;(C) = H X (S;2) 8 (E . It follows that
Tr(T*) + Tr(T*)
must
be the trace of an integer matrix and hence must be an integer itself. It's easy to see that many elements of Z[A] fail to satisfy this. Indeed, if we set A = {x e 7L[\] | x + x e 7L) then we see that ,2
A
—2
-A
A
,m
,...,A
is a free
.. —m
-A
, where
Suppose we let B are realizable. index.
2-module generated by
,
/
1 , A - A~
m = (p-l)/2
.
denote the group of elements of A
The question is:
Does
,
-i \ / <-»
which
B = A ? If not, what is the
Ewing: Automorphisms of surfaces
124
The G-Index Theorem The realizability question of the previous section seems, at first glance, to be hopeless. pute
Tr(T*)
It is extremely difficult to directly com-
for a given automorphism
T . The difficulty is not in
producing such automorphisms—we can produce them in great variety—but rather in determining the behavior of differentials.
T*
on the space of holomorphic
How then can we hope to compute all possible traces for
all possible automorphisms of a fixed period?
The answer is:
use the
G-Index Theorem or, more precisely, a special case of it known as the Eichler Trace Formula [3; p. 264]. The Index Theorem (without the thing:
G ) always says the same
the analytic index = the topological index.
It allows us to com-
pute certain analytic invariants in terms of topological information. The great strength of the theorem is that the indices take on many different meanings in many different contexts.
For example, in our special
context the Index Theorem says that the dimension of the space of holomorphic differentials (an analytic invariant) is precisely the genus of the surface (a topological invariant). The G-Index Theorem provides us with a similar tool when the situation we are studying comes equipped with an action of a group
G .
It allows us to compute the value of an analytically defined character of the group
G
in terms of topological information.
In our special case the value of the character is simply Tr(T*) . How do we determine it in terms of topological information? Suppose
T
is an automorphism of period
fix a (possibly empty!) set of isolated points ber that
T
P
p . Then , P ,...,P
T .
must be orientation preserving if it is holomorphic.)
will (RememWe
can describe the "type" of each fixed point by examining the differential of
T
acting on the tangent space at that point.
root of unity then
dT
k
tiplication by some power of A
If
acting on the tangent space at A , say
A
A
is a fixed p-th
P.
must be mul-
X
i . We say that
P.
has type
. The collection of fixed points together with their types is called
the "fixed point data" of
T .
Now the G-Index Theorem (in this special case) tells us how to determine is simply:
Tr(T*)
in terms of the fixed point data.
Indeed the formula
Ewing: Automorphisms of surfaces
125
1 1
Tr(T*)
P. A 1 X -1
(This is in fact a special case of the Atiyah-Bott fixed point formula [1; p. 459, ex. 1] which is in turn a special case of the G-Index Theorem.
It is, however, actually a result due to Eichler around 1930
and independently proved by Chevalley and Weil in 1934 [2]. A good reference for an "elementary" proof is [3; p. 256 ff.].) The G-Index Theorem allows us to do what seemed to be impossible:
We can compute
Tr(T*)
for an automorphism in terms of easily
determined, topological data. Realizing Fixed Point Data Our problem is nearly solved.
We must merely determine which
sets of fixed point data can be realized, and then solve the algebraic problem of determining which numbers of mula for
A
can be realized by the for-
Tr(T*) . Realizing fixed point data is a simple problem in covering
spaces.
Suppose we try to produce an automorphism
T k
surface
S
with fixed points
P ,... ,P
of types
X
of period k
i
p
on a
t
, ...,X
Our first observation is that it is enough to produce a diffeomorphism; it will be an automorphism for some complex structure. Now we also observe that after deleting the fixed points, is a p-fold covering of its orbit space and mation.
We can therefore try to build
with
deleted disks with boundary curves
t
S
T
from its orbit space. C
S
is just a deck transfor, C9,...,C
.
Start
(These
will be the deleted neighborhoods of our fixed points in the orbit space.) create
As usual, join the S
C.
by paths to a common base point.
To
we must first take a p-fold covering of this configuration—
such a covering is classified by a homomorphism group into
2Z/p/Z .
<j) from the fundamental
If we identify the deck transformation
generator for
Z / p Z , it is not hard to see that k. precisely when (KC. 1 ) = T .
P. X
T
as a k has type X
The key observation is that we can complete our construction of
S
and
union of
T p
if and only if the curve circles.
C
C2...C
lifts to a disjoint
We can then cap-off each circle and extend
T —
Ewing: Automorphisms of surfaces it simply permutes the caps. is the same thing,
126
This is true iff
<J)(C . ..C ) = 0
or, what
£ 1/k. = 0 mod p . This is a necessary and suffi-
cient condition for realizing a set of fixed point data. The remaining algebraic problem is straight-forward but messy. Recall that
A
(the set of all numbers in
2Z[A]
satisfying the obvious
necessary condition to be a trace) is a free Z-module generated by 1 , A - A""1,...,*111 - A~m , where index of the submodule
B
m = (p-l)/2 . We need to determine the
of numbers of the form
1-1
k
i
A
-1
where I 1/k. E 0 mod p . This is a familiar kind of problem to algebraists and simply requires finding (in principle) a basis for is then a determinant. nant.
B ; the index of
B
in
A
In this case it is, in fact, a familiar determi-
Indeed, it is the same determinant we mentioned in connection
with the class number: the index of
B
in
A
is
h
,
the first factor of the class number! Now the wonderful consequence is that all the work that number theorists have done for a century can be applied to our question about automorphisms.
For small primes
(< 23)h
= 1
and so
A
and
B
are
the same; there are no restrictions on which automorphisms can be realized other than the obvious ones. are quite far apart (recall that
For larger primes, however, h
~ 10
for
A
and
B
p = 199 ) ; there seem
to be many more restrictions on automorphisms of period
p —restric-
tions which are a little mysterious. Why is there any connection between the class number and automorphisms of Riemann surfaces?
Is it purely surperficial — an accident
of computation — or is there some deeper, underlying reason?
Unknown.
Ewing: Automorphisms of surfaces
127
For our purposes it is enough to point out that there JLS_ a connection and that it is made possible by the G-Index Theorem.
The
"significance" is that to answer our question we need not do any more work — the nubmer theorists have done it for us!
References [1]
Atiyah, M.F. and Bott, R. (1968). A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. of Math., 88, 451-491.
[2]
Chevalley, C. and Weil, A. (1934). Uber das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkorpers. Abh. aus dem Math. Sem. der Univ. Hamburg, JJ), 358-361.
[3]
Farkas, H.M. and Kra, I. (1980). Springer-Verlag.
Riemann Surfaces.
New York:
THE REAL DIMENSION OF A VECTOR BUNDLE AT THE PRIME TWO Edgar H. Brown, Jr. and Franklin P. Peterson
ABSTRACT Suppose £ is a real, stable vector bundle over a CW complex X of dimension less than 2n+l.
It is shown that the obstructions to realizing
5 as an n-plane bundle may be arranged so as to be a twisted Bockstein on w n , when n is even, and cosets of H*(X;Z2). INTRODUCTION In this paper we show that if £ is a real, stable vector bundle, then the obstructions to realizing £ by an n-plane bundle are 2-primary in dimensions less than 2n.
Under the assumption that E, is oriented, the
results of this paper were obtained by Copeland and Mahowald in [5]. See also the related results of Glover and Mislin [7]. Let f
B
k
(k)
°n —* BOn
Pk
—* B °
be the k*-" stage in the Postnikov system for the projection p: that is: W TT.(BO) «7T.(BO ), i £ in in
•K
V. (BO
in
J
) ~ 7T. (BO) , i >
l
Theorem 1,1. There is a tower of fibrations B M •> ... + B. + B. _ + ... B^ = BO
N
j
]-l
0
B 0 n -> BO,
Brown & Peterson: The real dimension of a vector bundle and a map F: 'Bjq -*• BOn (
i)
If p: B
129
such that
•*• B
is the composition of the above
projections, then p
F = p. (F is a 2 local
homotopy equivalence.) ( ii)
Except for n even and j = 1, B. •* B . ^ has fibre K(Z2,n.) .
(iii)
If n is even, B
+ B
has fibre K(Z
. ,n) and k-invariant
6*wn e H*(BO;Z^2 ) , where w R is the n ^ Stiefel-Whitney class, Z. . is the rational numbers with odd denominators, Z. . denotes local coefficients with respect to w, and 6* is the twisted Bockstein operation associated to Z
—• Z
. -*• Z
(see below for definition of 6 ) .
In [2] we constructed a space BO/I the projection BO/I
•> BO lifts to BO
over BO and conjectured that . ; this conjecture implies that
every smooth n-manifold immerses in R2n-a(n)m
Theorem 1.1 and the fol-
lowing theorem show that one need only consider obstructions in H*(BO/I ; Z2) . This fact is used by R. Cohen ( [4]) in his recent proof of the immersion conjecture. Theorem "1.2. If n-a(n) is even, then 6*w e H*(BO/I ; Z^_ is zero. t n-a(n) n (2) One may recast 1.1 in a more obstruction theoretic form) as follows: Corollary 1.3. Suppose £ is a k-plane bundle over a finite CW complex X with dim X < k, dim X < 2n, H q (X;Z 2 ) = 0 for q > n+1, and if n is odd, w
(Q = 0, or if n is even, 6*w (£) = 0 .
5' is an n-plane bundle and 0
Then 5 = ?' + 0
n
where
is the trivial (k-n)-plane bundle.
LOCALIZATION OF A MAP In this section we recall the definitions of a local system of groups, cohomology with local coefficients, the Serre spectral sequence with local coefficients, and localization of a map. Suppose X is a space.
Let P(X) be the category whose objects are
points of X and morphisms between x paths from x
and x. are the homotopy class of
to x . A local system of groups G, on X, is a covariant
Brown & Peterson: The real dimension of a vector bundle
functor from P(X) to the category of groups.
130
The cohomology groups,
H q (X;G) are constructed using cochains which assign to each q-simplex, T:
A
-> X, an element of G(T(d )) where d
is the 0
vertex of A
(see
[6]). Suppose F •> E ^-> B is a Hurewicz fibration with lifting function X, that i s , for each pair
(e,a) , e e E , a:
gives an element A(e,a) in p ' ^ a d ) )
I -* B , such that a(0) = p e , X
and X is continuous in e and a.
pose G is a local system of abelian groups on B. tem H^(F;G) as follows:
Sup-
We obtain a local sys-
for b e B ,
H q (F;G)(b) = H q ( p " 1 ( b ) , G ( b ) ) ;
and if a :
I -»- B
H q (F;G) (a) = G C O O ^ U * ) " " 1
p~ 1 (a(0)) •* p~ 1 (a(l)) is defined by X a (e) = X(e,a) .
where X a :
The orem 2,1. If G is a local system of groups on B and F •> E ^-> B is a Hurewicz fibration, then the Serre spectral sequence for H*(E;p*G) has its E_ term given by
Suppose p :
E -*• B is a fibration with simply connected fibre and
suppose
(k) is the Postkinov system of p . as follows:
(k-1) is constructed from E
let TT = TT (B) , TT = TT (B) , W the universal cover of K(TT,1) 1
and P
The space E JC
K
the space of paths in K(TT ,k+l) starting at the base point.
action of TT on TT gives an action of TT on K(TT ,k+l) and P . the induced fibration in
E(k)
K(
v
k)
i(k-D e
gv
k+i
^
X
TT P k
r +
W XTT
K(TT k ,k+l)
The (k) Then E is
Brown & Peterson: The real dimension of a vector bundle k+1 where 0 induces an isomorphism on TT ant lying in H
(E
131
and may be viewed as the k-invari-
;TT ) , the t indicating local coefficients.
Suppose R is a subring of the rationals, with unit.
We construct, by
induction, a tower of spaces and maps:
where the k H^l
fibre is K(TT ® R,k) and g induces an isomorphism, k k
I / T^
fi?k
n\
~ f*
Stl
T^ I
*^4
TT ~^1> I TT1
_ /""*
IV!
T^ 1
(k) for any local system G on E induced from a local system on B. Suppose k-1 k+1 k+1 ( (k—1) E and g _ have been constructed. Let 0 ® R e H (E ® R; TL ® R) correspond under g to j*0 e H (E ;TT ® R ) , where j:
k
JC"~1
K-
TT c TT ® R.
by 0
® R.
Then E ^
® R is the fibration over E (k-1) ® R induced
The existence of g
follows by construction and the fact
that it induces the appropriate cohomology isomorphism follows from 2.1 and H*(K(TT 8 R,h) ,G 8 R)) « H*(K(TT,h) ,G ® R ) for any group G. PROOF OF 1.1. Let F be the fibre of BO •> BO. n n BSO n -• BSO.
Note Fn is also the fibre of
We first consider the case n odd.
Then
Hq(BSO;Z[l/2]) x Hq(BSC^; Z [1/2]) for q < 2n+2, since the Pontrjagin class P
_.
is the first class in BSO
which vanishes on BSOn . Hence for i ~^ 2n. TT. (F ) is a finite ?qroup whose l nM ±order is a power of two.
The groups IT. (F R ) , i ^ 2n, are modules over
Z[TT. (BO)] = ztzo-'
modules, groups of order a power of two, are
nilpotent.
and
sucn
Hence by results in [1] , BO
sequence of fibrations with fibre K(Z o ,n.). ^ i = identity. This proves 1.1 for n odd.
->• BO may be factored into a We take B M = BO N n
n
and F
Brown & Peterson: The real dimension of a vector bundle Suppose n is even. n
fibre, namely, S . n < i ^2n.
and
Note F R •* ^n+1
B 0 n -»•
Bo
n+1
132
have the same
Hence TT. (F ) = Z for i = n and is a finite group for
In [3] it is shown that the first obstruction to a section
of BO -• BO, when n is even, is 6*w e H n t n twisted Bockstein defined as follows: v e H n (BO;Z 2 ), 6V = 2U, W
(BO; Zfc) , where 6. is the t
suppose V e d1 (BO;Z) represents
e C1(BO;Z) represents w
values 0 or 1 on each simplex.
Then, one defines 6
and W
only takes the
by
6 v = 6v - 2w u v. (One may easily check that this is the usual local coefficient boundary operator.)
Note
6 V = 2(U - W
u V) .
We define 6*v =* {U - W x U V } € H n + 1 ( B O ; Z t ) . Consider the following commutative diagram:
BO (k) (k) where BO and BO ® R come from B O •> BO as in section two, and all the n n n squares not containing B O are pullbacks.
The homotopy groups of the
fibres of p' and p' have order odd and a power of two, respectively. p.. :
Z -» Z[l/2] and p.:
Z •> Z
twisted Bockstein associated to w
are the inclusions and 6
If
is the
and Z -»- Z •> Z , then the k-invariants
of p^ and p 2 are P-t^^w and p 2 6*w n , respectively. To prove 1.1, n even, we take B = BO n ®Z._. . Then B + (n) N n (2) N BO ® Z(7) f a c t o r s i n t o fibrations with fibres K(Z 2 ,n.) as in the n odd case above.
Hence B
-> B O has the desired properties and it remains to
show that B O n + B has a section. n N ficient to show B^ -> B has a section.
Let B = BO
n
® Z / o . . It is suf(2)
Brown & Peterson: The real dimension of a vector bundle
We first show that P]_p{ has a section.
133
Consider the commutative
diagram: ®
Ztl/2]
Since n-1 is odd, TT. (F
) is a 2-group for i £ 2n-2 . Hence q is a homo-
topy equivalence and thus P-.P-! has a section. By construction Po<$*w
•^ .t n
p 6*w
= 0 e H*(B) .
= 0 in H*(B) and, since p., has a section, _
Suppose V e C
1
(B;Z) is a cochain representing
6. w : then t n
where V. e C
(B;Z) , q is odd and r is a power of two.
There are inte-
gers a and b so that aq + br = 1, and hence
Therefore 6, w = 0 in B and BO t n n
->• B has a section.
This section to-
gether with the section of p.. , from the previous paragraph, give two liftings of B -> BO to B •* BO n
® Z[l/2] which will differ by an element in
Hn(B;Z[l/2] ) . The Serre spectral sequence for K(Z
,n) -> B -> BO and
H*(B;Z[l/2]t) has, by 2.1, its E 2 given by E P ' q = H P BO;H q (K(Z ( 2 ) ,n.
But Hq(K(Z EP'
,n) ; Z [ l / 2 ] ) = 0 for q > 0.
= HP(BO;Z[1/2]
).
Hence E P ' q = 0 for q > 0 and
By the Thorn isomorphism theorem,
H P (BO;Z[l/2] t ) ~ H P (MO;Z[l/2]) = 0 . Hence H (B;Z[l/2] ) = 0, the two liftings above are the same and, since B •> BO ® Z[l/2] lifts to BO n n the proof of 1.1 is complete.
® Z[l/2], B. -• B has a section and 1
Brown & Peterson: The real dimension of a vector bundle
134
PROOF OF 1.2. Let MO/I
denote t h e Thorn spectrum of BO/I
H (MO/I ;Z 2 ) and U e H (MO/I ;Z ) .
•*• BO and l e t U e
The Thorn isomorphism for l o c a l coef-
f i c i e n t s has t h e form: U. u S t
Hq(BO/I ; Z t ] » Hq(MO/I ; Z) v n J n
where u. is defined as follows: t
if u,v e C*(BO/I ; Z) , n
u u. v(A ....,A ) = (-l)W!(AOAP)u(An,...,A )v(A ,...,A ) t 0 p+q 0 p p p+q We next show that (6*wk) u t U* = 6*(wk u U ) . One may check t h a t 6 ( u u t v ) = 6fcu u t v + ( - 1 ) ' U ' u Let p: Z •> Z-be the quotient map. Let u,v e C*(BO/I ; Z) represent w , ^ n K. and U , respectively. Then p(u u v) = p(u u v) and 6 v = 0. Hence 6*(wk u U) = {6(u u t v)/2} = {(6tu/2) u t v}
Let k = n - a(n) and suppose k is even.
To prove 1.2, namely, that
fc
6*w, = 0 in H*(BO/I ; Z ) , we may show that 6*(w. U) = 0 in H*(MO/I ; Z) . t K n K n Since w e I (see [2]), Sq (w U) = w U = 0. We show that w U = _ JC"J"X n Jc K.4"1 Jc Sq (y) and hence 6*(w U) = 0 because 6*Sq1 = 0. Let {u } be a basis for H*(MO;Z_) over A = Steenrod algebra. The set (x(Sq )u |l admissible}, x t n e canonical antiautomorphism of A, is a Z 2 basis for H*(MO;Z2) = H*(MO). H*(MO/I ) = H*(MO)/J n where J is the Z 2 subspace generated by x(s<3 )
u
where the first entry of
Brown & Peterson: The real dimension of a vector bundle I, i , satisfies 2i w U = Sq k U = I and Sq x(Sc3. ^)U e J.
> n - |u |.
Hence
x(Sq 3)U If I. has 1 as its last entry, Sq X(Sc5 "') = 0 and
X(Sq 3) = Sq X(ScI ) • Otherwise, Sq X(s<3 •*) = x(ScI J.
135
^'
)
and
Hence w, U = Sq y and the proof of 1.2 is complete. Brandeis University Massachusetts Institute of Technology REFERENCES
1. A.K. Bousfield and D.M. Kan; Homotopy limits, completions and localizations, Springer-Verlag, Lecture Notes in Mathematics 304 (1972). 2. E.H. Brown and F.P. Peterson; "A universal space for normal bundles of n-manifolds", Comment. Math. Helv. S4 (1979), 405-430. 3. N. Steeiirod; The Topology of Fibre Bundles, Princeton University Press, 1951. 4. R. Cohen; "The immersion conjecture for differentiable manifolds", to appear. 5. A.H. Copeland, Jr., and M. Mahowald; "The odd primary obstructions to finding a section in a V bundle are zero", Proc. A.M.S. 19 n (1968), 1270-1272 . 6. E. Spanier; Algebraic Topology, McGraw Hill, 1966. 7. H.H. Glover and G. Mislin; "Immersions in the metastable range and 2-localization", Proc. AJM.S. _29 (1971), 190-196. NOTES 1.
The work in this paper was supported by NSF grants.
MAPS BETWEEN CLASSIFYING SPACES, III J.F. Adams and Z. Mahmud
§1.
Introduction.
Let G and G 1 be compact connected Lie groups, and
let f: BG — > BG 1 be a map.
It is shown in [2] that the induced homo-
morphism f*: K(BG) < — K ( B G ' ) carries the representation ring R(G') c K ( B G ' ) into the representation ring R(G) c K(BG).
Moreover the
induced map R(G) < — R(G') can also be induced by a homomorphism 0: T — > T 1 , where T,T' are the maximal tori in G,G'.
Here of course
one has to state that the behaviour of 8 with respect to the Weyl groups W , W is such that 6*: R(T) < — R (T') does indeed carry R(G') C R ( T ' ) into R(G) c R(T); in [2] maps 0
with this behaviour are called "admissible
maps". Our present purpose is to see what further information can be obtained by using real and symplectic K-theory.
Let RO(G) <= R(G) be the
subgroup generated by real representations; similarly for RSp(G) c R ( G ) , using symplectic representations.
Assume that the group G is semi-
simple . Proposition 1.1.
For any map f: BG — > BG 1 , the induced homo-
morphism R(G) < — R ( G ' ) preserves real elements, in the sense that it carries RO(G') into RO(G); similarly it preserves symplectic elements, in the sense that it carries RSp (G1) into RSp(G). The main problem, however, is to take this result and deduce useful, explicit conclusions about 6: T — > T 1 .
For this purpose we must recall
Adams & Mahmud: Maps between classifying spaces, III some representation-theory.
137
In particular, if p is any irreducible
(complex) representation of G, then under p any element z in the centre Z of G acts as a scalar.
If z
2
= 1 in G, then of course the scalar p(z) is
Lemma 1.2 . (after Dynkin [5]). Lie group.
(a) Let G be a compact connected
Then there is a canonical element 6 e Z <= G with the following 2
properties, (i) 6
=1.
(ii) For any irreducible self-conjugate repre-
sentation p of Q,
p(6) is +1 or -1 according as p is real or symplectic.
(b) More explicitly, when G is simple 6 is as follows.
If G is SU(n)
with n odd, Spin(m) with m = 0,1,2 or 7 mod 8, G 2 , F^, E. or EQ then 6=1.
If G is SU(n) with n even or Sp(n) then 6 is the matrix -1. If
G is Spin(m) with m = 3,4,5 or 6 mod 8 then 6 is the non-trivial element in the kernel of Spin(m) — >
SO (m) .
non-trivial element in the centre Z. 4 2 centre such that £
= 1 / then 6£
If G is E- then 6 is the unique (c) If C c Z is any element of the
also has properties (i) and (ii) of
part (a). We shall write I c z for the subgroup of elements £ , where 4 C e Z and £ = 1 ; and we shall regard I as an "indeterminacy" which affects elements satisfying (i) and (ii) of part (a). Our object is now to conclude that if we take f: BG — > BG1 and pass to an associated admissible map 0: T — > T 1 , then 06 = 6' mod I 1 ; in other words, 0 preserves the Dynkin element (modulo indeterminacy). It is apparent from Lemma 1.2 that this condition is sufficient for 0*: R(G) < — R ( G ' ) to preserve real and symplectic elements; we aim to prove that this condition is also necessary, at least in certain cases. This calls for two comments.
First, we wish to emphasise that
the condition "06 = 6 1 mod I"1 is indeed useful and explicit.
For
example, consider the case G = G1 = Sp(l); the centre Z = Z 1 is {+1}, and
Adams & Mahmud: Maps between classifying spaces, III
138
the Dynkin element 6 = 61 is -1. The appropriate maps 9 are those of the form 9 (t) = t , so the condition "06 = 61 mod I 1 " becomes "k is odd". Secondly we wish to emphasise that further assumptions will be needed.
For example, in the case G = G 1 = Sp(l) we know [6,7] that the
correct conclusion is "k is odd or_ zero", and we know that the case k = 0 can actually occur (take the map f: BG — > BG 1 to be constant at the base-point).
So in this case we shall need some assumption to ex-
clude the case k = 0. In fact, in general 06 need not even lie in the centre Z 1 of G 1 . (For an example, consider the usual injection Sp(n) — > Sp(n+1) .)
How-
ever, we have the following result. Proposition 1.3.
Let f: BG — > BG1 correspond to an admissible
map 0: T — > T 1 which is irreducible in the sense of [2] p20; then 0 carries Z into Z 1 . Somewhat weaker assumptions should suffice to prove that 06 lies in Z 1 ; for it is sufficient to verify that 06 lies in the kernel of each root of G 1 , and apparently this could be done by methods similar to those given below.
However, as the case of an irreducible admissible
map is the most interesting one, (1.3) is probably enough. For our main result, we shall make the assumptions which follow, (i) G1 is one of the simply-connected classical groups Spin (m), SU(n), Sp(n). (ii) 0: T — > T 1 is an admissible map. (iii) 0*: R(G) < — R ( G ' ) preserves real and symplectic elements, (iv) 06 e Z 1 . (v) If G 1 = Spin(2n) with 2n = 0 mod 4, or if G' = Sp(n), we assume that the map 0 is irreducible in the sense of [2] p20.
Adams & Mahmud: Maps between classifying spaces, III To state our final assumption we need some notation. x1,x2,...,x
be the basic weights in G
1
139 Let
= Spin(2n), Spin(2n+1), SU (n) or
Sp(n); and set 6. = 6*x., so that the map 6 has components 0. (except in the case G 1 = SU(n), in which we have a relation ^0. = 0 ) .
Fix a Weyl
chamber C in G, and let T € W be the element which carries C to -C. (vi) If G' = Spin(m) with m ± 2 mod 4 or G' = SU (n) with n = 2 mod 4 we assume that no 0. is fixed by T. Theorem 1.4.
Under these conditions we have B6 = 6l mod I 1 .
We pause to comment on these assumptions.
In (i), the only
exceptional group we need to exclude is E 7 , which would involve ad hoc calculations.
We presume that for E- it would be appropriate to assume
that 0 is irreducible. ever assumption
Assumptions (ii) to (iv) seem acceptable.
How-
(iii) can sometimes be weakened, as we proceed to show.
Lemma (1.2) (b) yields many cases in which 61 £ I 1 , so that every selfcon jugate representation of G 1 is real; in such cases 0 automatically preserves symplectic elements, and so it is enough to assume that 0 preserves real elements.
The opposite happens when G 1 has enough basic
symplectic representations.
For example, if G 1 = Sp(n) then it is
sufficient to assume that 0*A
is symplectic.
k
For if so, then since 0*
k
commutes with A , we see that 0*X
is real or symplectic according to the
parity of k, and similarly when A
is replaced by a monomial
(A ) ^•(A2) 2 ...(A n ) n .
Inspection of the proof of (1.4) will show what
is actually used in each case. Assumption (v) serves to rule out cases like the inclusion SU(n) c Spin(2n) for n = 0 mod 2 and the inclusion SU(n) c Sp(n) for n = 1 mod 2; in these cases the conclusion fails, although all the other assumptions hold.
Assumption (vi) serves to rule out cases like the 2i exterior powers A : SU(n) — > SU(m) where m = n!/(2i)I(n-2i)I and
Adams & Mahmud: Maps between classifying spaces, III m = 2 mod 4 (for example X
: SU(4) — > SU(6)).
140
In these cases also the
conclusion fails, although all the other assumptions hold.
Assumption
(vi) is not too restrictive; for many groups G we have T = -1, and then 6^ can be fixed by x only if 0. = O - a possibility which is normally ruled out when 6 is irreducible.
In §5 we shall see how one can make use
of assumption (vi). The remainder of this paper is organised as follows. prove Proposition 1.1.
In §2 we
In §3 we deduce Lemma 1.2 from the statement
originally given by Dynkin; we also prove Corollary 3.1, a useful result of representation-theory which however did not need to be stated in the introduction.
In §4 we prove Proposition 1.3, and in §5 we prove the
main result, Theorem 1.4. The first author thanks the University of Kuwait for hospitality during the preparation of this paper. §2.
Preservation of real and symplectic elements.
In this section we
will prove Proposition 1.1. We begin by studying the case in which the group G is simplyconnected.
In this case one of the main theorems of Hermann Weyl
Cl pl64] asserts that the representation ring R(G) is a polynomial algebra; and the proof of the theorem gives a preferred set of generators. More precisely, the weights in the closure of the fundamental dual Weyl chamber form a free (commutative) semigroup, with a unique set of generators, say 03 ,O32,. .. ,co
[1 pl63]; to these weights there correspond
irreducible representations p ,p2,...,p ; and the theorem says that R(G) is a polynomial algebra Z[p,p 2 ,...,p.] on these generators. The map -1: L(T) — > L(T) is conjugate to a map -T (see §1) which preserves the fundamental Weyl chamber C.
It follows that -T
Adams & Mahmud: Maps between classifying spaces, III
141
permutes co ,u)2, .. . ,w . Thus complex conjugation permutes the irreducible representations p.^p-,...,?..
Those of p.,p2,...,p
which are self-
conjugate are either real or symplectic, but not both [1 p64]. Complex conjugation also permutes the monomials i
Pi
l
i
±
2
Po
mials.
£
Po •
Let us
restrict attention to the self-conjugate mono-
We will call such a monomial "real" or "symplectic" according as
the sum of the exponents of symplectic generators p. is even or odd. Since a representation of the form pp is real [1 pl66], it follows that each "real" monomial is a real representation, and similarly each "symplectic" monomial is a symplectic representation. Lemma 2.1.
The subgroup RO(G) of R(G) has a base consisting of
the following elements. (i) m, where m runs over the real monomials, (ii) m+m, where (m,m) runs over pairs of distinct conjugate monomials. (iii) 2m, where m runs over the symplectic monomials. The subgroup RSp (G) of R(G) has a base consisting of the following elements. (i) 2m, where m runs over the real monomials, (ii) m+m, where (m,m) runs over pairs of distinct conjugate monomials. (iii) m, where m runs over the symplectic monomials. Proof.
In view of the discussion preceding the lemma, it is
clear that the subgroup generated by the elements listed is contained in RO (G) or RSp(G) as the case may be. For the converse, we recall that it is easy to prove the corresponding descriptions of RO (G) and RSp (G) in terms of real irreducible representations, pairs of distinct complex conjugate irreducible
Adams & Mahmud: Maps between classifying spaces, III
142
representations, and symplectic irreducible representations; see for example Cl p66]. Any irreducible representation p can be written as a Z-linear combination of monomials, p = £ a m , and therefore p+p can be written as p+p = £ a (m +m ) . Consider now a monomial m = p
p 2 ...p
Let a be the irreducible representation corresponding to the "highest weight" i-. co +i200o+. . .+iooo ; then a occurs in m with multiplicity 1 •L 1
^ ^-
[1 ppl61-164].
36 36
Thus m is self-conjugate or not with a, and in the/self-
conjugate case, m is real or symplectic with a.
It now follows that
when we write a self-conjugate irreducible representation a 1 by induction as a Z-linear combination of monomials
m\
we use, apart from sums
m+in, only monomials m" which are real or symplectic with a 1 .
This
proves the lemma. We now introduce the reduced generators a. = p.-(ep.)l, where e is the augmentation. I (RG).
These generators lie in the augmentation ideal
We still have R(G) = Z[a ,a 2 ,...,a.].
Complex conjugation
permutes ai'ao'•••'ao J u s t as it permuted p ,p 2 ,..., p ; a. is selfconjugate or not with p.? and in the self-conjugate case, a. is real or symplectic with p..
We may form monomials a.. a 9 . ..a0
\ l2
is self-conjugate or not with p case, we call a 1
a 9 ...a ^
Lemma 2.2.
p.
; such a monomial
h p
; and in the self-conjugate
"real" or "symplectic" with p 36
p 1
. ..p0 • 2.
36
The description of RO (G) and RSp(G) in Lemma 2.1
remains valid if we use monomials m = a. a 9 •••0
in the reduced
generators. This follows immediately from Lemma 2.1. Next we recall that by theorems of Atiyah and Segal [3 pplO, 14,17] / we have canonical isomorphisms
Adams & Mahmud: Maps between classifying spaces, III
R(G) A
—-—>
K(BG)
RO(G) A
—-—>
KO(BG)
RSp (G) A Here R (G)
>
143
KSp (BG) .
means the completion of R (G) with respect to the topology
defined by powers of the augmentation ideal I (RG).
The topologies on
RO(G), RSp(G) may be taken to be the restrictions of the topology on R(G); see [3 pl7], but note that the reference there to (5.1) should be to (6.1).
The following diagrams commute.
R0
(G)A
—Z.—>
Z
KO(BG)
R(G) A
—
RSp (G) A
—*—>
KSp (BG)
R(G) A
—*
K(BG)
>
>
K(BG)
We proceed to describe these completions explicitly in our case. Corollary 2.3.
R(G)
is the ring of formal power-series
zCCa.,ao,..., a ]]. An element of R(G) formal Z-linear combination \ a m m
= a
a 2 • ••
(a
may be written uniquely as a e Z) of monomials
Such an element is self-conjugate if and only if
conjugate monomials appear with equal coefficients.
If self-conjugate,
it lies in the completion RO (G) A of RO (G) if and only if the coefficient of each symplectic monomial is even; it lies in the completion RSp(G) of RSp(G) if and only if the coefficient of each real monomial is even. This follows immediately from the work above.
Adams & Mahmud: Maps between classifying spaces, III Corollary 2.4.
144
In K(BG) we have
R(G)nKO(BG) = RO(G) and R(G)nKSp(BG)= RSp(G). Proof.
Take a typical element of K(BG) = R (G)
Z-linear combination £ the coefficients a
a
T
m
i
#
I f i-t
l
are non-zero.
ies in
as a formal
R(G) only finitely many of
If it lies in KO(BG) - RO(G)
then
conjugate monomials occur with equal coefficients, and the coefficient of each symplectic monomial is even.
If the element lies both in R(G) and
in RO(G) , then it lies in RO(G).
The proof of the second statement is
similar. Proof of Proposition 1.1.
By Corollary 1.13 of [2], an induced
map £*: K(BG) <
K(BG')
carries R(G') into R (G) ; it clearly carries KO(BG') into KO(BG), and KSp(BG') into KSp(BG).
So when G is simply-connected, Proposition 1.1
follows from Corollary 2.4. If G is merely semi-simple, let G be its universal cover, so that we have a finite covering map IT: G — > G.
Then the preceding result
applies to the composite BQ
—?IL_> BG
_> BG. #
So from x e RO(G') we infer (B7r) *f*x e RO (G) . From this it follows that f*x £ RO(G).
If we assume x e RSp(G'), a similar argument applies.
This
proves Proposition 1.1. §3.
Dynkin elements.
We begin by recalling the work of Dynkin [5]; the
reader may also consult Bourbaki [4], These authors assume that G is semi-simple, so we make this assumption for the moment and remove it
Adams & Mahmud: Maps between classifying spaces, III later.
Then the construction given for 6 is as follows.
145
Choose a Weyl
chamber C; this determines a base of simple roots <j) ,cj> , .. . ,
consider these roots as linear maps L(T) — > R.
2.
36
Then there is a unique
vector t e L(T) such that <J>. (t) = 1 for i = 1,2,...,£.
It follows that
every root takes an integer value on t; so t maps to the identity under the adjoint representation, and t yields an element 6 of the centre. The authors cited guarantee the behaviour of p(6) in the case they consider; we will discuss the other assertions of Lemma 1.2. The element 6 constructed above appears to depend on the choice of C; however, any other Weyl chamber C 1 may be obtained from C by the action of an element w e W, which will carry 6(C) onto 6 (C'). fixes central elements, 6 (C) = 6(C')f
Since W
and so the construction is
canonical. In particular, -C is another Weyl chamber; it leads to simple ^ —1 2 roots -({>-,-(|)2,...,-cj) and to the vector -t. Hence 6 = 6 and 6 = 1 . This establishes assertions (i) and (ii) of the lemma when G is semi-simple.
Moreover, the construction of 6 yields the following
further properties.
(iii) If G is a product group G-iX G o ' then the Dynkin
element 6 in G is the product of the Dynkin elements ^ w ^ o
in G
-i'G2*
(iv) Let G — > G be a finite covering; then the Dynkin element 6 in G maps to the Dynkin element 6 in G.
If we insist on preserving these two
properties, and define the Dynkin element of any torus to be 1, we get a canonical extension of 6 from the class of compact connected semi-simple groups to the class of compact connected groups;
and this extension has
the properties (i),(ii) stated in the lemma. Part (b) lists the value of 6 in each simple Lie group, and this is an easy calculation.
Adams & Mahmud: Maps between classifying spaces, III It remains to prove part (c) , about &£, .
Since 6
146
= 1 and we
4 assume C = 1* and since 6 and £ are both central, it is clear that (6C )
=1.
Suppose then that p is an irreducible self-conjugate repre-
sentation of G, as in the lemma.
Since p is self-conjugate we get
= P (?) , so p (C ) = 1 and p (6£ ) = p (6) . This
p(C) = p(C), that is p(C)
completes the proof of Lemma 1.2. Next we recall that weights a) may be considered as 1-dimensional representations of T, and symmetric sums S (oo) as representations of T. We note that a central element z e Z acts as a
scalar under S (oo) ; in
fact, since W fixes central elements we get (caw) z = oo(wz) = oo(z) for each w. We shall call a self-conjugate symmetric sum S (co) "real" or "symplectic" according as S (co) 6 is +1 or -1. Corollary 3.1.
The subgroup RO(G) of R(G) has a base consisting
of the following elements. (i) S (oo) where S (oo) runs over the real symmetric sums, (ii) S (oo)+S (oo) , where (S (oo) ,S (oo)) runs over pairs of distinct conjugates. (iii) 2S(oo), where S (oo) runs over the symplectic symmetric sums. The subgroup RSp(G) of R(G) has a base consisting of the following elements. (i) 2S (oo) , where S (oo) runs over the real symmetric sums, (ii) S (oo)+S (oo) , where (S (oo) ,S (oo) ) runs over pairs of distinct conjugates. (iii) S (oo) where S (oo) runs over the symplectic symmetric sums. If we were allowed to replace the symmetric sum S (oo) by the irreducible representation p (oo) with oo as an extreme weight, this would become a standard result of representation-theory; see [1] p66.
We can
Adams & Mahmud: Maps between classifying spaces, III
147
write each p (co) in terms of the S(o)1), and (by induction over oo) each 5 (to) in terms of the p(oj'); the work above assures us that in this process we only use weights u)' with a fixed value of GO1 (<5) • §4.
Preservation of the centre.
as the universal cover of T.
We interpret the Stiefel diagram L(T)
As in [2 p22], let F be the extended Weyl
group of G (generated by the reflections in all the planes of the Stiefel diagram, not just those which pass through the origin); and let T
be the
subgroup of translations in T. Lemma 4.1.
Let f: BG — > BG1 correspond to 8: T — > T'; then
6 = L(0) carries r into r . o o Proof,
—•
f induces a map from TT. (G) = TTO (BG) to IT. (G1) = TT O (BG') . 1
A
1
2.
1
Here we can suppose without loss of generality that IT (G ) is free abelian, for we can arrange this by passing to finite covers of G and G 1 a step which does not change 6 = L(6). The diagram = TT2(BT)
7T
is now
1(G)
>
= 7r (BG) 2
*
TT2(BT') = IT
7r
2(BG')
=
^i* 6 '*
commutative, for when TT (G1) is free abelian this follows from
the results on rational cohomology in [2]. We can interpret T Q as the i
k e r n e l o f TT (T) —> TT_ (G) , and r a s t h e k e r n e l of IT. (T') —> 1 1 o 1 ~ » h e n c e 6 = L(6) c a r r i e s V i n t o r . o
o
Proof of Proposition 1.3. to a vector z e
TT. (G 1 ) ; 1
Take an element z e Z, and lift it
T = L(T). By Proposition 2.28 of [2] (applied with G'
replaced by G) we have
wz E z mod r
o
Adams & Mahmud: Maps between classifying spaces, III for all w € W.
148
Let a: W — > W 1 be as in Theorem 2.29 of [2]; applying
0 = L(6) and using Lemma 4.1, we get (aw) (S'z ) = (0"z ) mod r . Since in Proposition 1.3 we assume 0 irreducible, Theorem 2.29 (ii) of [2] yields € Z1.
? z So 0 carries Z into Z'.
§5.
Location of self-conjugate symmetric sums.
Theorem 1.4 is as follows.
The pattern of proof of
We first select some irreducible self-
conjugate representations p. of G 1 .
We take care to choose enough
representations p. so that the conditions z1 e Z1,
( z 1 ) 2 = 1 and p!(z') = 1 for all i
imply z 1 e I 1 ; we shall see that this can be done in each case.
It is
then sufficient to prove that p . (06) = p . (61) for each i. Since we assume that 0* preserves real and symplectic elements, 0*p. is real or symplectic with p..
The crucial step is now to prove
that 0*p . contains at least one self-con jugate symmetric sum S(o).) with odd multiplicity.
If so, then we can apply Corollary 3.1 to the ex-
pression for 0*p. in terms of symmetric sums, and conclude that S(w.) is real or symplectic with 0*p. and p.; that is, S((O (<5) = p^(6') . i
But since S(o).) occurs in 0*p . , we also have
Adams & Mahmud: Maps between classifying spaces, III
149
Thus
and this completes the proof. To fill in this outline, we must first show that we can find enough representations p..
Let us dismiss as trivial the cases
G1 = Spin(n) with n = 2 mod 4 G' = SU(n)
and
with n f 2 mod 4.
In these cases the conditions z1 e Z \
(z1)2 = 1
already imply z' e I', so we need choose no representations p.. Next we take the cases Spin(n) with n = 1 mod 2 SU(n)
with n = 2 mod 4
and
Sp (n) . In each case we have just 2 elements z 1 € Z 1 such that (z')
2
= 1 , and
i
one representation p. will do.
It is sufficient to take the spin-
representation A on Spin(n) with n = 1 mod 2, the exterior power A SU(2m) where 2m = 2 mod 4, and the fundamental representation A
on
on
Sp(n) . Finally we take the case Spin(n) with n = 0 mod 4. In this case the centre is Z 2 X Z?•
It
^ s sufficient to take two repre-
sentations p., namely the fundamental representation A , and either one of the two half-spin representations A , A . This completes the choice of the p..
Adams & Mahmud: Maps between classifying spaces, III
150
The essential step is now to show in each case that 6*p. contains at least one self-con jugate symmetric sum S(u).) with odd multiplicity.
We consider first the cases Spin(2n) with 2n = O mod 4 Sp(n)
in which we have to consider p. = A •
In both cases W' permutes the
2n weights +X-./+x2, ... ,+x ; therefore W permutes the 2n elements — 0 1'— 6 2''' " — 9 n ' and they fall into orbits under W.
If (cf> ,<j)2,... ,<{>r)
is an orbit, then (-<j> ,-2, . ..,-cj> ) is also an orbit.
Suppose to begin
with that one of the orbits other than (<j> ,<j>2, ... ,<j> ) has the same elements as (-<(>_, -2,.. ., -<J> ) . Then we can factor the admissible map 0; if G 1 = Sp(n) we use the subgroup U (r) *Sp(n-r) , while if G 1 = Spin(2n) we use the pull back in the following diagram. -> Spin(2n)
U(r)*so(2n-2r)
> SO(2r)xSO(2n-2r)
> SO(2n)
This contradicts the assumption that 9 is irreducible; therefore, it does not happen.
We conclude (firstly) that (-<{> r-^o' ••• /"• ) i s
tne
same
orbit as <4>,<J>2 ) ; therefore, each orbit has the form (+4> ,-Hj; ,. .. ,-Hj; ) . We also conclude that no orbit is repeated, for if the orbit (<{>,/2, ...,<J>r) were repeated, the second copy of (<j> ,<|)2,. .. ,<\> ) would have the same elements as (-(j> ,-<{>2, .. .,- ) . This says that when we decompose 6*A
into symmetric sums, corresponding to
the orbits (j^ ,+ij; ,. .. ,-HJJ ) , each symmetric sum is self-conjugate and occurs with multiplicity 1. In the remaining cases we have to consider p. either A
or A~.
We use the same argument in all cases.
= A, A Let
and
Adams & Mahmud: Maps between classifying spaces, III
1 1 1 (i) ± J 6 !'! 2 ^ 2 ' " " - 2^m
in the cases G
'
=
s
151
Pin(2m)
G 1 = Spin(2m+1) (ii) 61,eo,...,0- in the case G 1 = SU(2m). 1 2 2m In either case we have 2m
) . = o .
By assumption (vi) , <J> . is not fixed by T , SO the condition (f> . ( T - 1 ) V / o is satisfied by an open dense set of vectors v e L (T) .
It
follows that we can find a vector v such that (f> . ( T - 1 ) V f O for all j. Now,
T permutes (j) , <j) , .. . ,(j>
, and for the element § . T we have
=-4). (T-l)v. So the elements ,2, . .. ,<J>
fall into pairs (<J),<J>T), with one member of
each pair being positive on ( T - 1 ) V and one member of each pair being negative.
Without loss of generality, we can suppose the <J>'s renumbered
so that <{>,({>,... ,§
are positive on (T-1)V.
We claim that
0). = <J).+<J>«+•••+(() is one of the weights of 9*p . . case G1 = SU(2m), 6..
p. = X
This is clear in the
since OJ. is merely the sum of some m of the
In the cases G 1 = Spin (2m)
and Spin(2m+1) the (j)' s already fall into
pairs — z 1
— z Z
— 2m
taking opposite values on (x-l)v; we must have selected one from each pair and obtained ( e e + e 6 + + £ 6 )
where each e. is +1.
If G = Spin(2m+1) this is one of the weights of
Adams & Mahmud: Maps between classifying spaces, III
9*A, because 2"( £ 1 x 1 + e 2 x 2 + * * * + e m X ^ then a),
is a w e i
9ht
is either one of the weights of 0*A
of A
*
152
If G = Spin(2m)
or one of the weights of
e*A~. Next we claim that this weight occurs with multiplicity 1 in 9*p..
In fact, among the weights of 9*p . , oo. is by construction the one
with the maximum value at (x-l)v . Finally we claim that the symmetric sum S(oo.) is self-conjugate. In fact, by construction we have
-(1+<j>2+.. .+m) ( s i n c e
2m
£ <J> . = 0)
This proves the required result for 9*A and for 0*Xm; and for G 1 = Spin(n) with n = 0 mod 4 it shows that the required result holds either for 6*A
or for 0*A . This is enough, and it completes the proof.
References [1]
J.F. Adams, "Lectures on Lie Groups", W.A. Benjamin 1969; to be reprinted by the University of Chicago Press.
[2]
J.F. Adams and Z. Mahmud, "Maps between classifying spaces", Inventiones Math. 35 (1976) 1-41.
[3]
M.F. Atiyah and G.B. Segal, "Equivariant K-theory and completion", Jour. Differential Geometry 3 (1969) 1-18.
[4]
N. Bourbaki, "Groups et algebras de Lie" Chap.VIII (especially pages 131-133), Hermann, Paris 1975.
[5]
E.B. Dynkin, "Maximal subgroups of the classical groups", in American Math. Soc. translations, series 2, volume 6, American Math. Soc. 1957, 245-378.
[6]
J. Hubbuck, "Homotopy homomorphisms of Lie groups",in'New developments in topology", L.M.S. Lecture Note Series no.11, C.U.P. 1974, 33-41, especially pages 33-34.
Adams & Mahmud: Maps between classifying spaces, III References (cont.) [7]
Z. Mahmud, "The maps BSp(l) — > BSp(n)", Proc. Amer. Math. Soc. 52 (1975) 473-478.
153
IDEMPOTENT CODENSITY MONADS AND THE PROFINITE COMPLETION OlF TOPOLOGICAL GROUPS
A. Deleanu Syracuse University, Syracuse, New York, U.S.A.
The profinite completion defined on the category of groups [11] , [12] presents the difficulty that it is not idempotent, that is, the iterated profinite completion is not equivalent to the single one [1], [8]. By developing an idea of Frank Adams [1], [2], it is shown in this paper that this difficulty can be avoided by defining profinite completion as a functor on the category of topological groups. The framework of codensity monads is used in order to describe various profinite completion functors, and a general criterion for such a monad to be idempotent is established. Idempotent monads were discussed by Peter Hilton, Armin Frei and the author in [7]. 1 Idempotent codensity monads Recall that a monad (or triple) on a category C [9] consists of a functor T : £~»C and two natural 2 transformations T| : 1 •* T , p, : T -> T which make the following diagrams commute: 3
T
2
>T >
I* T
1
,
T
=
=
T
T
.
A monad < T , T), p, > is called idempotent if 2 71TX : TX-»T X is an isomorphism for all X € C . Plainly, < T , Tj, p. > is idempotent if and only if p. is a natural equivalence, PROPOSITION 1,1. The monad < T , T|, p, > is idempotent if and only if T]T = TT| .
Deleanu: Idempotent codensity monads
155
Proof, Assume is idempotent. Since p, • TIT = 1,= p, • TT) and p, is a natural equivalence, we have T]T = p, = TT|. Conversely, suppose T|T = TT| • The naturality of T] yields the commutative diagram
iT 2 But from our assumption we infer TjT = TT|T , so that TIT •p, = T p , • TIT = T p , •TT]T = T ( p , • T)T) = T ( l
t h i s , combined with
p. •T]T = 1_ / shows t h a t
) =1
p, i s an
9
T
;
equivalence. Now, let K : M-»C be a functor such that the right Kan extension R:£->C o_f K along K exists. Thus [9] , we have a natural transformation e : RK -* K such that, for any functor S : C-*C , the assignment a »* e • aK is a bijection cp : NatCS.R) =Nat(SK,K) . We set S = 1^, and define T] : 1 _ -• R by T] = cp" (1T.) ; we then set S = R" and define p, : R z -* R by p, =. cp""x(€ • Re ) . It is easy to check that Tj and p, are natural, and that KZ' those maps JL : Z -»Z' in M for which (Kj&)f=f; there is a projection functor Q v : (XiK)-»M sending f to Z . Then RX=limKQ v # and, if h : X •* Y in C , Rh is the unique <- X — morphism in C which makes the diagram XX RX = limKQ v 3^—> KZ
RhJ RY = litnKQ
a
> KZ
Deleanu: Idempotent codensity monads
156
X Y (resp. \ ) i s commute for every g 2 Y -»KZ in (Y|K), where \ the l i m i t i n g (universal) cone of limKQv (resp. limKQ__). The d e s c r i p t i o n of
T] in terms of l i m i t s i s given
by PROPOSITION 1.2. unique morphism in
For each
X €C , l e t
a x be the
C which makes the diagram RX = lim KQY (1) KZ = KQx(f)
commute for every
f € (X1K) .
Then
a : 1 K -• R is a natural
transformation and a = T| . Proof. in C .
To check the naturality of a , let h : X -> Y
Then, in view of the above description of Rh , we can
write for every g € (YlK) X Y Y X (Rh)av = v \ a Y = gh = X*ar h , g x gn x g Y which implies (Rh)a x =a h . To prove that that is e •CTK= 1 K .
a = T) , we must show that cp(a) = 1 ,
Taking
X = KZ
and f = 1 z
in diagram
X* av- = 1 _ . But, by [9, p. 234], X^ Z = e . 1 X KZ K Z ^ KZ Z We shall need in the sequel the following
(1), we get
z
DEFINITION [10, p. 118]. in a category
Let u : A -> B b e a morphism
D and let C be an object of D . u is said to
be epimorphic relative to C if D(u,C) : D(B,C) -» D(A,C) is an injective map. THEOREM 1.3.
The codensity monad
K : M-»C is idempotent if and only if, for each X 6 C , 1^ is epimorphic relative to KZ for every Proof. Proposition 1.2, diagram
Z€ M.
First, note that for each X € C , by T]
is determined by the condition that the
Deleanu: Idempotent codensity monads
157
(2)
commutes for every
g € (RXIK) , whereas, by the above descrip-
tion in terms of limits, R(T]V) is determined by the condition X. that the diagram
RX^
*•'
commutes for every
(3)
g € (RXjK) .
Now, assume that, for each X € C, 7]__ is epimorphic X. relative to KZ
for every
Z €M.
This implies, since by
Proposition 1.2 there is for each g € (RXjK) a commutative diagram
that
X ~ = g . Then diagrams (2) and (3) show that giix T)_v = R (T)__), so that, by Proposition 1.1,
Then, by Proposition 1.1/
7] = R(T] ) for each X € C . This RX x implies, in view of diagrams (2) and (3), that Xg I ~1 = g for
each
g € (RXiK) . h
RX
x
Thus, if two morphisms .
KZ
are given such that hTl = X Tl , we have
h= so that
T)
^hTk = X*\ = Z '
is epimorphic relative to KZ
for each
Z €M
Deleanu: Idempotent codensity monads
158
2 Examples of codensity monads Throughout the following examples, K will be the appropriate inclusion functor. (1) Let C be the category of groups and M the full subcategory of finite groups. Then R is the profinite completion of groups [11], [12]. This monad is not idempotent, as shown by the example of a direct sum of an infinite number of cyclic groups of order p , where p is a fixed prime [1], [8]. (2) Let C be the category of groups and M the full subcategory of A-nilpotent groups [5], where A is a solid ring. Then R is the A-completion in the sense of Bousfield and Kan [5, p.103]. This monad is not idempotent, as shown by the example of a free group on a countably infinite set of generators, with A = ZZ [4, p.57]. (3) Let C be the homotopy category of pointed connected CW-complexes and M the full subcategory of those which have finite homotopy groups. Then R is the Sullivan profinite completion [1], [11], [12], denoted by Su in [6]. This monad is not idempotent, as shown by the example of an Eilenberg-MacLane space corresponding to the group mentioned in example (1) above [1]. (4) Let £ be the category of topoloqized objects of the homotopy category of pointed connected CW-complexes, as defined in [1], [2], [6], and M the full subcategory of those topologized objects whose underlying CW-complexes have finite homotopy groups. Then R is a generalized profinite T completion, denoted by Su in [6]. The question as to whether this monad is idempotent is open. (5) Let C be the category of topological groups and M the full subcategory of finite groups with discrete topology. Then R is the profinite completion of topological groups. This monad is idempotent, as shown in Theorem 3.1 below. REMARK. More generally, throughout the preceding examples, we can replace "finite groups" by "finite P-groups,
Deleanu: Idempotent codensity monads
159
where P is an arbitrary family of primes." We thus obtain P-profinite completion functors. 3
The profinite completion of topological groups
THEOREM 3.1. Let C be the category of topoloqical groups and continuous homomorphisms, and let M be the full subcategory whose objects are the finite groups with the discrete topology. Then the codensity monad (R,TJ,M,> of the inclusion functor K:M-ȣ is idempotent. Proof. First, for each topological group G , consider the family {N } of all closed normal subgroups of G of finite index, and denote by J the subcategory of (GiK) whose objects are the canonical projections onto quotient groups p
: G
and whose morphisms are the canonical homomorphisms qotp -» G/N whenever Na c Npo , o : G/N ' a ' po Each such subgroup is also open, so that G/N is indeed a discrete finite group. We show that the inclusion functor I : J-» (GJK) is initial [9, p.214]. First, given an arbitrary object of (G1K), f : G-»KF, there is a commutative diagram in C :erf) G T
iKs
and ker f is plainly a closed normal subgroup of G of finite index. Second, given a commutative diagram in C p
K(G/Na) KF
N HNp is a closed normal subgroup of G of finite index, so
160
Deleanu: Idempotent codensity monads
that N O N = N., for some index Oi
p
Y , and we have the commuta-
Y
tive diagram in (: K(G/N a ) s
Now. since
I is i n i t i a l and limKQ_I obviously *•
G
exists, by the dual of Theorem 1 on p.213 of [9] limKQ also exists and there is a canonical isomorphism RG = l i m K Q ^ — — > limKQ^I = limK(G/N ) such that
v
for each
co = X
p 6 J # where
v is the
limiting cone of limKQ I .
If we put G=limK(G/N ) and e^=u>^TL# then, for 4-
Oi
'
G
G G G
each topological group G ,
is the unique map such that p
:G-»K(G/N ) .
v
0 =p
for each
For, by Proposition 1.2,
Next, we show that the image of 0_ is dense in G . Q ^
————
To this end, let x be an arbitrary point in G , and let 0 be an arbitrary open subset of G such that x 6 0 . x = (x ) , where Oi
x € G/N '
Oi
for each
Then
a , and 0 = V n G , where V
Oi
is an open subset of n(G/N ) . Thus an open subset of G / N for each
x 6 n U c V oi , where U n
a ot and U^Now = G/N^
is is a Pi N Nalli but for
a finite set of indices, a-,,. closed normal subgroup ofsay G of finite index, so that n H N =N for some index 3 . Let y € G be such that
p R (y) = x . P
Thus
P
Then p (y) a
(y) =
for
0 =
v ^ G _ ( y ) = p (y)= x for a=a,,...,a , so that p G a oi in
Deleanu: Idempotent codensity monads
0
(y) €IIU
.
161
Hence 9^(y) € 0 , and x belongs to the closure
G a ot G of the image of 9 . The fact that the image of 9 is dense in G implies that 9 , and therefore also TlQ , is epimorphic relative to any Hausdorff topological group, in particular relative to KF for every discrete finite group F. But according to Theorem 1.3 this means that (R,T],p,) is idempotent, which concludes the proof of the theorem. Thus, for every topological group G , we have the profinite completion of G : RG = G = limK(G/N ) . We can describe this completion in terms of completions of uniform spaces, in a manner which is similar to the p-adic completion of the additive group of integers, as follows: Take the family [N } of closed normal subgroups of G of finite index as a fundamental system of neighbourhoods of the identity element, thereby defining a second topology on the group underlying G / which is compatible with the group structure; let G1 denote this new topological group. Being closed and of finite index, each N is also open in G, so that the homomorphism i : G->G' defined by i(x) = x for all x € G is continuous. This also implies that {N } coincides with the family of closed normal subgroups of G1 of finite index. By considering the uniform space structure associated to G' , we can form the Hausdorff completion (T1 of G1 , which comes equipped with a canonical continuous homomorphism \|f : G1 -»G' [3, p. 248]. The existence of G1 is guaranteed by Corollary 2 on p.290 of [3], which also yields the existence of an isomorphism T ' in £ such that the diagram
Deleanu: Idempotent codensity monads
162
commutes. But each N is open in both topological groups G and G / so that both G/N and G'/N are discrete. Consequently limK(G/N ) and limK(G'/N )a r e isomorphic qua 1
topological groups, and we conclude that there exists an isomorphism T in £ such that the diagram
commutes. Remark. The author is very grateful to the referee for pointing out the papers of Lambek and Rattray [14], [15], [16] , [17] ; their work starts with a codensity monad on a complete category and then uses a method due to Fakir [13] in order to turn it into an "idempotent monad. They give extensive applications to localizations and completions. [1] [2] [3] [4] [5] [6] [7] [8]
References Adams, J.F. (1975). Localization and completion. Lecture Notes in Mathematics, University of Chicago. Adams# J.F. (1975). Adams' Problems. In ManifoldsTokyo 1973, University of Tokyo Press, Tokyo, pp. 430-431. Bourbaki, N. (1966). Elements of Mathematicso General Topology, Part 1/ Hermann, Paris. Bousfield, A.K. (1977). Homological localization towers for groups and n-modules. Memoirs of the Amer. Math. Soc. ±2, no. 186. Bousfield, A.K. and Kan, D.M. (1972). Homotopy limits, completions, and localizations. Lecture Notes in Mathematics No. 304, Springer-Verlag, Berlin. Deleanu, A. (1982). Topologized objects in categories and the Sullivan profinite completion. Journal of Pure and Applied Algebra ,25, pp. 21-24. Deleanu, A., Frei, A. and Hilton, P. (1975). Idempotent triples and completion. Math. Zeitschrift 143,
pp. 91-104. Huber, M. and Warfield, R.B. (1982).
p-adic and p -
cotorsion completions of nilpotent groups. of Algebra 7j4, pp. 402-442.
Journal
[9] Mac Lane, S. (1971). Categories for the working mathematician. Springer-Verlag, Berlin. [10] Pareigis, B. (1970). Categories and functors. Academic Press, New York.
Deleanu: Idempotent codensity monads
163
[11] Sullivan/ D. (June 1970, revised April 1971). Geometric topology. Part I: localization, periodicity and Galois symmetry. M.I.T. Notes. [12] Sullivan, D. (1974). Genetics of homotopy theory and the Adams conjecture. Annals of Math. 100, pp. 1-80. [13] Fakir, S. (1970). Monade idempotente associee a une monade. C.R. Acad. Sci. Paris 270, pp. A99-A101. [14] Lambek, J. (1972). Localization and completion. Journal of Pure and Applied Algebra 2L» PP« 343-370. [15] Lambek, J. and Rattray, B.A. (1973). Localization at injectives in complete categories. Proc. Amer. Math. Soc. 41, pp. 1-9. [16] Lambek, J. and Rattray, B.A. (1974). Localization and codensity triples. Comm. in Algebra _1/ PP- 145-164. [17] Lambek, J. and Rattray, B.A. (1975). Localization and duality in additive categories. Houston J. of Math. JL, pp. 87-100.
FINITARY AUTOMORPHISMS AND INTEGRAL HOMOLOGY J. Roitberg Hunter College, CUNY, New York, N. Y., U.S.A.
1_. The notion of finitary automorphism (or pseudo-identity) of a group was introduced by J. M. Cohen in [C]. We recall the definition. Definition.
An automorphism
each element such that
x € G
<$>: G •* G
of a group
G
is finitary if
lies in a finitely generated subgroup
K = K
of G
<J> restricts to an automorphism <j>|K: K = K Obviously, any automorphism of a finitely generated group G
is automatically finitary.
The following two examples of nonfinitary
automorphisms should serve to illuminate the concept. Example 1. r
Let V be an arbitrary nontrivial group and let G = © V n=—°°
denoting a 'copy' of F,
morphism, carrying each T
n G Z.
If
isomorphically onto T
,
auto-
_ , then cj> is a
nonfinitary automorphism of G. Example 2.
Let G be the additive group of dyadic rationals, consisting
of those fractions having denominator a power of 2.
If <J>: G •* G is
'multiplication by 2 ' then (j) is a nonfinitary automorphism of G. In [C], Cohen, whose sole concern was with abelian groups G, established a spectral sequence result with applications to fibrations of simply-connected spaces.
Motivated partially by the desire to extend
Cohen's result from the simply-connected to the quasi-nilpotent case ([HRl]), the authors in [HR2] develop the theory of finitary automorphisms
of locally nilpotent
groups
G
and in
[CHR]
of finitary automorphisms in the homotopy category.
initiate a theory The following two
results set the stage for the problem to be studied in the present paper; the first is Theorem 6 of [HR2], the second is a special case of one of the main theorems in [CHR] Theorem 1.
If
$: G + G
concerning nilpotent spaces.
is an endomorphism (not assumed to be an auto-
morphism) of a nilpotent group, then $ is a finitary automorphism ° the induced endomorphism ,: G , -•• G , on the abelianization of G is a * ab ab ab finitary automorphism.
Roitberg: Finitary automorphisms and integral homology Theorem 2.
If <J>: G -»• G
165
is an endomorphism of a nilpotent group, then
<j> is a finitary automorphism
°
the induced endomorphism (j>#: H^G -> H^G
on the integral homology of G is a finitary automorphism. With regard to Theorem 1, it is easily seen that the implication ** is valid for an arbitrary group G.
The converse implication <•
fails for general G since being a finitary automorphism does not ab guarantee that <j> is an automorphism. Taking T group (that is, T
= 1) and (J>: G -* G
to be a nontrivial perfect
as in Example 1, we see that even
if <{> is an automorphism, it may not be finitary merely because <J> is ab finitary. In a different vein, it may be noted that even if G is nilpotent,
need not be an automorphism merely because <j> . is an autoab morphism. In fact, let N be a nonabelian group, N < 0 i f n nilpotent
rn = •
N ,
G =
0
n=-oo
F
n
and $: G -> G
if n > 0
the obvious shift map.
Plainly G is nilpotent,
<J> is not an automorphism but <J> . is an automorphism. Of course, <J> . is ab ~ ab the type of automorphism met in Example 1 and is not finitary. As for Theorem 2, again the implication ** fails for general G; indeed, we may reason as above for Theorem 1, replacing 'perfect' by 'acyclic' (cf. [BDH]).
In contrast to the situation noted above with
respect to Theorem 1, however, we remark that for nilpotent G, must be an automorphism if <J># is an automorphism. For the remainder of the paper we focus attention on the question of the validity of the implication =* in Theorem 2 for general G. We have: Theorem.
There exists a finitely presented group
4>: G -> G (necessarily finitary)
G
and an automorphism
such that <j>^: H^G -v H^G
is not a
finitary automorphism. The proof of this theorem together with some additional commentary will occupy the next section. It is a great pleasure for me to dedicate this paper to Peter Hilton on the occasion of his sixtieth birthday.
The observations
contained herein are, in part, an outgrowth of work resulting from our collaboration, a collaboration which I trust will continue for many years to come.
Roitberg: Finitary automorphisms and integral homology
166
2_, We show that the group G constructed by Stallings in [S] supports an automorphism of the desired kind. Following the notation in [S], we begin with a free group on two generators A = . element a.
If B is the
have the presentation
Let cj> : A -*• A
B = ,
ated by the indicated elements. morphism <j> of B.
denote conjugation by the
normal subgroup of A
generated by b,
then we
that is, B is freely gener-
We note that <j> restricts to an auto-
Hence, if C is the qeneralized free square A * A ,
B <J> naturally induces a unique automorphism A C =
B <J> : C -* C; writing C
= a«ba2 , n £ Z> ,
<J> is determined by the assignment a. —>- a
, a o —>• a 0 , b —>- a ,ba n
(= a ? b a« ) . The next step is to embed C in the finitely presented group D =
= a 1 ba 1
=
a
2
ba
2
>
and to observe that if <J> : D ->• D denotes conjugation by the element x, then D|c = <J> . Finally, we take G to be the generalized free square D * D and let § - \» : G •> G be the unique automorphism induced by <j> . We claim that 4>^: H G ->• H G
is a nonfinitary automorphism.
To this end, we begin by observing that ( ) : B -> B , and note that ((()_) , is precisely the autoB at> ai) aJD B ab morphism discussed in Example 1 with T = Z. Now (<j) ) : H A -v H A is A -L 1 certainly a finitary automorphism (actually, (6 ) . = identitv since cf)_ \
is an inner automorphism).
*
•"
A
By applying [HR2; Corollary 3] to the map
of Mayer-Vietoris sequences H c
^
H2C
^
H;LB
H^^B
we conclude that ( ) ^ : H C •> H C Finally,
^n^*
: H
o
D
H
"*" 9
D
^ H A e H.A
> HXA e HXA is a nonfinitary
^
s a f n:
automorphism.
tar
i "-
Y automorphism (again, (<j> ) ^
= identity), so by once again applying [HR2; Corollary 3] to the map of Mayer-Vietoris sequences
Roitberg: Finitary automorphisms and integral homology
167
(V*e V * we conclude that (J)^: H G ->• H G is a nonfinitary automorphism, thereby completing the proof of the theorem. We have seen that the automorphism <j>^: H~G -»» H G manufactured from the type of Example 1.
is suitably
nonfinitary automorphism presented in
As was kindly pointed out to me by Gilbert Baumslag, it is
also possible to begin from the type of
nonfinitary automorphism
presented in Example 2 and fashion a suitable example. We very briefly outline the construction. Take A =
= b >, with <J> conjugation by the
element a. Let B be the normal subgroup of A generated by b. It is readily checked that B is isomorphic rationals
and that <j>
to the additive group
restricts to an automorphism <j>B of B
be identified with 'multiplication by 2 ' . and 6 ,
1
of dyadic which may
The constructions of C, D, G
are then performed in precisely the same manner as
example.
Since it is known ([BDH]) that there is a functorial embedding of any finitely generated group
G
into
a finitely generated
acyclic group A G , it follows that any finitely generated group G admits a finitely generated 'suspension' EG = AG * AG. A S this E construction G is likewise functorial, any automorphism <j): G •*• G induces a unique automorphism
Ecf>: EG -* EG
and a simple Mayer-Vietoris argument shows
that (J)^ is finitary <* (£)*
is finitary.
suspension operation to the (j>: G -* G
By applying the iterated
of the theorem (or even to <J) : O
C),
we thus easily deduce the following corollary to the theorem. Corollary.
There exists a finitely generated group K
of arbitrarily
large homological connectivity and an automorphism i\n K -v K i|^: H^K -v H^K
such that
is not a finitary automorphism.
If it were known that there is a functorial embedding of any finitely presented group into a finitely presented acyclic group, then we could, of course, strengthen the corollary suitably.
There is a
process (cf. [BDM]) for embedding a finitely presented group into a finitely presented acyclic group but this process is definitely not a functorial one.
Roitberg: Finitary automorphisms and integral homology To close, we wish to raise a further problem.
168
The various
groups considered in this section appear to be rather complicated in many ways.
It would seem fair to say, for instance, that they are very far
from being nilpotent.
Thus, it might be interesting to find an example
validating the theorem with the group G, say, a solvable group. In [BD], the authors construct a pair of finitely presented metabelian groups, each with infinitely generated integral homology groups in most degrees, so these groups are perhaps good candidates. Unfortunately, it is not at all clear how to concoct a suitable automorphism on either of these groups.
Bibliography [BD]
Baumslag, G. & Dyer, E. (1982), The integral homology of finitely generated metabelian groups, I. Amer. J. Math. 104, 173-182.
[BDH] Baumslag, G., Dyer, E. & Heller, A. (1980). The topology of discrete groups". J. Pure & Appl. Algebra 16^ 1-47. [BDM] Baumslag, G., Dyer, E. & Miller, C. F. On the integral homology of finitely presented groups. To appear in Topology; see also (1981) Bull. Amer. Math. Soc. 4_, 321-324. [CHR] Castellet, M., Hilton, P. & Roitberg, J. In preparation. [C]
On pseudo-identities II.
Cohen, J. M. (1968). A spectral sequence automorphism theorem; applications to fibre spaces and stable homotopy. Topology 1_, 173-177. ' "" ~
IHR1] Hilton, P. & Roitberg, J. (1976). On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations. Quart. J. Math. 27_, 433-444. [HR2] Hilton, P. & Roitberg, J. [S]
On pseudo-identities
I.
To appear.
Stallings, J. (1963). A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85, 541-543.
FINITE GROUP ACTIONS ON GRASSMANN Henry H. Glover
and
MANIFOLDS
Guido Mislin
Ohio State University and ETH Zurich to VdtiK Wilton on tkz o£ hi& bixtizth birthday)
occasion
INTRODUCTION. Let the finite group
G.
then by applying
a
automorphism by
p
p : G -*• GL (3C) denote a complex representation of n. If
a G Gal(E/£)
is a field automorphism of E
to the entries of a matrix one obtains an induced
o^ : GL (CC) -*• GL ((E) . We denote the composite map a^ o p
and we call it a Galois conjugate of
two representations
p
and
p
p.
are equivalent
Since
G
is finite,
(p ^ P )
if and only O
if their characters and
T
x
an
d
X
act in the same way on Let
CC to
if
a
y.
^ p
deg (a) = k mod m,
m
a natural number, Note that
is surjective and Aut(y) ^ lim Aut(y ) , —
m-th roots of unity.
•*-
p
(equivalence class) of the tensor product representation of
p
p : G -* GL
the Grassmann manifold we write and
p
(E (p) s, t
{C s, t
(CC) of
is a representation, then s-dimensional
for the corresponding
are projectively equivalent
y c. y m m
As usual, we denote by
If
if a
for the map given by restricting an automorphism
m-th roots of unity by the k-power map.
deg : Gal(CC/D) -> Aut(y) the group of
p
|G|-th roots of unity in (E .
We use the notation
acts on
In particular
y C (C be the group of roots of unity and write
deg : Gal(<X/$)) ->• Aut(y) of
agree.
T
G
® p
the
and
p .
acts on
linear subspaces of
G-manifold.
Clearly, if
(C p
1
(i.e., if there exists a one dimen-
;
Glover & Mislin: Finite group actions
sional representation
* A : G ->• CC
such that
p
170
^ A Q& p0) ,
there is an
equivariant homeomorphism
CC (p ) ^ CC (p ) . If p~ denotes the comS, t i S/t 2 plex-conjugate of the representation p, then complex conjugation induces an equivariant homeomorphism
CC (p) ^ CC (p) . Liulevicius proved [16] s, t s, t that conversely if there is an equivariant map CC (p ) -> CC (po) i S,t 1 S,t 2
which is a homotopy equivalence on the underlying space, then there exists a one dimensional representation
A
such that either
p
^ A® p
or
We will study more generally the connection between equivariant maps
f : (C
(Pn) -*• CC
(p )
and the representations
p
S / t l S f t 2
and
1
p ; 2
Liulevicius1 result will correspond to the case where
f
equivalence.
for which one needs
We will also deal with the case
s = t,
is a homotopy
Hoffman's result [12] on the automorphisms of the cohomology algebra of CC (cf. Lemma 1.5). Our results should be compared with similar res, s suits on finite group action on a sphere (see Atiyah-Tall [2] and LeeWasserman [14]).
The proofs are very different however; we follow essen-
tially the method initiated by Liulevicius in [16]. It is convenient to use the following notion of degree for a self map integer
f : CC -> CC . Since H (CC ; 7Z ) ^ 2Z , there is a unique S/t S/t S/t deg(f) corresponding to the induced homomorphism of this second
homology group.
The Kahler class, which generates
maximal cup length; thus a map of degree
± 1
H (CC ; 7L ) , has s, t
is necessarily a homotopy
equivalence (cf. Brewster-Homer [5]). Our main result can be stated as follows. Main Theorem: P. / Py ' G -* GL
Let
s
and
t
be natural numbers and let
(CC) be representations of the finite group
G.
Suppose
Glover & Mislin: Finite group actions there exists an equivariant map
f : C S/t
to
|G|.
171
(p ) -> C (p ) X S , t 2.
of degree prime
Then A)
There exists a one dimensional representation such that
Bl)
p
and
X& p
are Galois conjugate.
a E Aut((C/$)
denotes an automorphism such that
p
then
^ X® p .
If
s ^ t,
or if
a
and
s = t
in cohomology (i.e.
X
f and
of
G
If
are related as follows. f
induces a grading map
f x = deg(f) x
for
x E H
) ,
then
deg(a) = deg(f) mod |G|. B2)
If
s = t
and
mology, then The maps
f
f
does not induce a grading map in coho-
deg(a) = - deg(f) mod |G|.
considered in our Main Theorem are in general non-
linear in the sense that they are not induced from linear maps of a linear
f
has necessarily degree If the representation
defined over
$,
then
p ^ p
0 p
or
CC
;
1.
is equivalent to a representation
for all
a E Gal(£C/$).
This is for in-
stance the case for all representations of symmetric groups.
The follow-
ing is then an immediate consequence of our Main Theorem. Corollary: map between Grassmann degree of
f
Let
f:£C
s,t
(p.) -*• (C (p.) 1 s,t 2
G-manifolds where
is prime to
G
is a symmetric group.
|G| then there is a linear
Our paper is organized as follows. Main Theorem.
If the
G-homeomorphism
In section one we prove the
We show in section two how to construct equivariant maps
in the case of actions on jective space
be an equivariant
(CP .
En ,, that is, n-dimensional complex prol,n+l It is somewhat mysterious how to construct such equi-
Glover & Mislin: Finite group actions variant maps in the general situation.
172
However, it is possible to formu-
late our Main Theorem in a more general way (section three). has then a converse, stating that if then there is a map
p
The theorem
is Galois conjugate to
f : (E (p ) •> (C (po) S,t 1 S,t 2.
of degree prime to
p , |G| ,
which is homotopic to an equivariant map at "each .prime which divides |G|"
(cf. Theorem 3.3). LIULEVICIUS1 USE OF REPRESENTATION
1.
We will write of the compact Lie group
K.
R(K)
for the complex representation ring
The Adams operations ty : R(K) -* R(K)
then defined in the usual way. a 6 Gal(JC/5D)
RINGS.
If
G
is a finite group,
gives rise to an automorphism
from mapping a representation
p
well known relationship between
to i|/
p . and
are
then
R(a) : R(G) •*- R(G) ,
induced
The following lemma states the R(a)
(a proof may be found in
Eckmann-Mislin [7]). Lemma 1.1. prime to
|G ,
If
G
then for every
denotes a finite group and a & Gal((C/$)
such that
k
is an integer
deg(a) = k mod|G|
one has ty = R(a) : R(G) -> R(G) . Let
p : G -> U(s + t)
be a unitary representation of the
* finite group
G.
Then
R(U(s + t)) - algebra. turns
R(U(s) x U(t))
p
: R(U(s + t)) -* R(G)
makes
Similarly, the usual inclusion into an
R(U(s + t)) - algebra.
R(G)
into an
U(s) x U(t) + U(s+t) Thus we can form
the tensor product RG & which we will denote by an obvious
R(U(s) x u(t)) R(U(s + t))
(RG<&R(U(s) x u(t))) .
R(G) - algebra structure.
This tensor product has
Glover & Misiin: Finite group actions To solve our problem on Grassmann forth assume that all representations
p
173
G-manifolds, we will hence-
of
G,
which we consider, are
unitary; there is of course no loss in generality in doing so. p : G -> U(s + t) E
s /1
is such a representation, then the usual identification
^ U(s + t)/U(s) x u(t)
K-theory of
If
C
(p)
allows us to compute the equivariant
as follows (compare Snaith [18] and Liulevicius [16]).
S/t Lemma 1.2. p : G -> U(s + t) ,
If the finite group
then there is a natural
G
acts on
via s, t R(G) - algebra isomorphism
A(p) : (R(G) \2> R(U(S) *u(t))) p If
d> : (RG®R(U(s) *U(t)))
E
+ K°((E (p)). u s,t
->(RG0R(U(s) *U(t))) Pi
a map of
RG - algebras then we define the degree of
the augmentation
dim : RG -> 5Z
is P2
<j> as follows.
and tensor ing with
Using
Q) gives an induced
map f : $S»
Since from
$&
R(U(s) x U(t)) ->$<£> R(U(s) x U(t)) R(U(s + t)) R(U(s + t))
R(U(s) x U(t)) R(u(s + t))
K° ((E S
_ <{>, using the Chern character, an endomorphism
•
H ((E ; {£) . s, t
we obtain
* <j>
of
<J> in general to be the degree of the composite of
* 2 * (J) : H ((C ;$)-»- H (I ; $) s, t s, t ; (B) -> H2((E S , U
) Q> Q '
This endomorphism might fail to preserve the gradation; we
define the degree of
H*«C
is isomorphic to
with the projection
; Q) . S, u
The following three lemmas are crucial for the proof of our theorem.
They will permit us to reduce the degree
degree one situation as considered in [16].
k
situation to the
Glover & Mislin: Finite group actions Lemma 1.3. finite group k G 7L
G
Let
and let
is such that
p : G -> U(s + t)
a 6 Gal((E/$)
174
be a representation of the
be a Galois automorphism.
deg(a) = k mod |G| then there is a map of
If RG -
algebras 0, : (RG&R(U(s) x U(t))) -> (RG(2>R(U(s) * U(t))) k p pa of degree
k, * 0
ing map
given by in
H (E
K
Proof.
a®$->-a®ip3.
Moreover,
0
induces a grad-
* ; $).
S ,t
The map
0 k
is induced from the following diagram, whose rows
have as push-outs the algebras in question:
R(G) < ( P
M Note that unit mod
k
}
R(U(s + t))
> R(U(s) x u(t))
| . j/
R(G) <
|/
R(U(s + t))
is necessarily prime to
> R(U(s) x U(t)). |G| since the degree of
a
is a
|G|. The commutativity of the diagram follows from Lemma 1.1
and the naturality of the 0" : Q) ®
^-operations.
The induced map
R(U(s) x U(t)) + $ Of R(U(s) x U(t)) R(U(s + t)) R(U(s + t))
corresponds under the natural isomorphism $ ®
to the Adams operation
ty
R(U(s) x U(t)) ^K°((C ) S> Q S R(U(s + t)) ' on vector bundles, which correspond via the
Chern character to a grading map of degree well known.
Thus
0
has degree
k
k
on
H((C
; $ ) / asis s, t and induces a grading map 0 in
Glover & Mislin: Finite group actions Lemma 1.4.
The map
0
Q) <& 9
: $ <2> (RG <E> R(U(s) x u(t) ))
Proof.
It suffices to show that
k
175
of Lemma 1.3 induces an isomorphism -> $ © (RG <S> R(U (s)
$& 0
x U(t))) .
is injective, since its domain
k
and range are finite dimensional (the dimension is T E Gal((C/$) m E 7L
^-vector spaces of the same dimension
N • dim ($ & RG)
such that
is such that
where
N = (s + t)!/(s!t!)).
deg(cfT) = 1 mod |G| and thus
deg(x) = m mod |G| then
(p )
Choose
^ p.
If
$08
= ($ 0 0 ) ($ <E> 0 ) mk m k i s a n e n d o m o r p h i s m o f Q <8> ( R G 0 R ( U ( s ) x u ( t ) ) ) ^ K° (C ( p ) ) <2>Q. B y p ~ G s,t Slominska [17], K° «E (p)) C£> Q injects into II (K° (I (p)H) &> RH<S>$) G s, t HC G s, t and for every component of the map into the product one has a commutative
diagram 5) © (RG ® R ( U ( s )
xu(t)))
> $ <S> RH <^) K ° ( I ( p ) H ) s,t
P
(D (E> 0 * mk Q (S> (RG 0
R(U(s)
x u(t) ) )
> $ 2) RH (S> K° (C P
But
mk 1 ® \\> : $ 0
K° (CC
s,t
(p)
H
(p)") s,t
) -> $ <2) K° ((E
s,t
(p)
H
)
is obviously
an
iso-
morphism, as one can see by passing to ordinary cohomology with rational coefficients, using the Chern character. so is
Thus,
Q) (S) 0 . mk
is monic and
© ® 0, . k Next, we will describe the structure of endomorphisms and auto-
morphisms of the cohomology algebra of
(C which we will need. s, t
corresponding results are scattered around in the literature.
The
Early re-
sults are due to Glover-Homer [10]. Brewster [4] classified all endomorphisms of non-zero degree of
H ((E ; 7L ) if s ^ t; his method applies s, t equally well to the cohomology with coefficients in a field of characteristic 0. The automorphisms of H (I ; 7L ) were determined by s, s
Glover & Mislin: Finite group actions
176
Hoffman in [12]; again, his methods yield a corresponding result in case of coefficients in a field of characteristic
0.
A discussion of the
automorphism conjecture for generalized flag manifolds and its connection to rational homotopy theory is presented in Glover-Mislin
[11].
For our
Main Theorem w e need the endomorphism result for rational coefficients; this is discussed entirely in Brewster-Homer
[ 5 ] . W e will also need the
corresponding result for coefficients in the field of (cf. section t h r e e ) .
p-adic numbers
The sources mentioned above imply the following
general result.
Lemma 1.5. *
Let
F
*
<j> : H (CE ; F) -> H (CC ; F) s,t s ,1 k G F,
k ^ 0. (i)
Then If
b e a field of characteristic be an
0
and let
F-algebra endomorphism of degree
<J> is an automorphism and the following holds. s ?* t,
4>
is a grading map (i.e.,
<j>x = k x
for
x 6 H2n). (ii)
If
s = t,
m a p , where
then either K
*
*
* (j> o K
is a grading
*
: H (CC ; F) -> H (CC ; F) s, s s, s
involution of degree s-plane in
<J> or
CC
-1
denotes the
induced from mapping an
to the orthogonal complement with
respect to some fixed unitary metric on CC Proposition 1.6. variant map of degree map
k
deg a = ek mod |G|, where £ = - 1
is commutative
f : CC (p ) -»• CC (p ) S,t I S,t 2.
prime to
f : CC (p ) -*• CC (p ) S,t L s,t 2.
mology, and
Let
and
be an equi-
|G|. Then there exists an equivariant a G Gal(CC/$)
e = + 1
if
f
such that
induces a grading map in coho-
in the other case, such that the following diagram
Glover & Mislin: Finite group actions
1
(® <8>e r
177
o f1
(RG (2) R(U(s) x U(t))) P2
> $ <£> (RG (S> R(U(s) x U(t))) a Pi dim
%> dim © 1 id $ <£
R(U(s) x U(t)) R(U(s + t))
Proof.
If
f
> $<2>
induces a grading map, we take
and 1.4 respectively.
If
f
f = f
f © K
K : (C (p ) -> (E (p ) s,s 1 s,s 1
is a grading map (of degree
(C
.
case by taking
- k).
to be equivariant, by mapping an
s-plane to the orthogonal complement with respect to a metric on
and apply Lemma 1.3
does not induce a grading map, we are in
case (ii) of Lemma 1.5 and thus We can choose
R(U(s) x U(t)) R(U(s + t)
p -invariant
The assertion of our Proposition then follows in this f = f © K
and noting that
deg(K) = - 1.
The proof of our Main Theorem can now be completed by applying Liulevicius1 result [16], From there it follows that the existence of a diagram as considered in Proposition 1.6 implies that there is a one di* mensional representation
X : G -> CC
deg(a) H deg(f) mod |G| if
f
a such that
Similarly,
deg(a) = - deg(f) mod |G| if
f
2.
^ A® p .
induces a grading map,
Theorem follows by Lemma 1.5.
can only occur in case
p
B2
Bl
Since
of the Main
follows by noting that
does not induce a grading map, which
s = t.
EQUIVARIANT MAPS ON PROJECTIVE SPACES. We will discuss the construction of equivariant maps as
considered in the Main Theorem for the case of
(C
= £CP ,
complex
1 , IITI
projective
n-space. Lemma 2.1.
finite group
G.
Let
Assume that
p : G -> GL ((C) be a representation of a n+1 p
is a sum of representations which are
Glover & Mislin: Finite group actions
178
induced from one dimensional representations of subgroups of a 6 Gal({C/$) k G 7L ,
is a Galois automorphism and if
deg(a) E
G.
If
k mod |G|,
then there exists an equivariant map f : (CPn(p) + EP n (p Q )
of degree Proof.
k.
Since
p
is a sum of representations induced from one dimensional
representations, there exists a basis
{en , ..., e . } 1 n+1
of
(E
such
that with respect to this basis
P(9)(Z
where and
1
h_ , ... , h G (E and 1 n+1 h, , ..., h , 1 n+1
rily
|G|-th
| G| — th.
(h
l
Z
h
*(l)
n+l ^(n+l)'
IT a permutation (cf. Dornhoff [6]); j\
all depend on
roots of unity.
g G G,
Since
a
and
h, , ..., h , 1 n+1
acts by the
are necessa-
k-power map on
roots of unity, we infer p (g)(z
°
Let
=
W
P
denote the map
induced map
P f
= (h
W (C
l %
-*- C
given by
a k k p (g) o P = P o p(g)
by construction
degree of
l
= f : (DP (p) ->• (CP (p ) is obviously equal to
for all
h
n + l \(n+l) ) -
Z z.e . I g G G.
> Z z. e. .
Then,
Therefore, the
is an equivariant map, and the
k.
Using this Lemma, we obtain the following partial converse to our Main Theorem. Theorem 2.2.
Let
p , p o : G ->• GL . (CC) 1 2 n+1
tions of a finite nilpotent group and
* X : G -»• (C
G.
be two representa-
Suppose there exists
a G Gal((C/$)
a such that
p
^ A & p . Then, for every
k = deg(a) mod |G|, there exists an equivariant map
k G 2Z
with
Glover & Mislin: Finite group actions
with Proof.
179
deg(f) = k. Since
G
is nilpotent,
p
is equivalent to a sum of represen-
tations induced from one dimensional representations of subgroups [6]. By Lemma 2.1 we can therefore construct an equivariant map g : (CP (p ) + (CP (p )
of degree
equivariant homeomorphism f = h o g
k.
Since
We consider G
G - CW
there is an
of degree one.
Thus
k.
EQUIVARIANT MAPS AT A PRIME
Illman [13]. If
^ X^ p ,
h : (CP (p ) ->• (CP (p )
is the desired map of degree 3.
p
p.
complexes
X
in the sense of
is a finite group, then the localization and com-
pletion of Bousfield-Kan [3] may be used to define functorially localized G - CW
complexes
X
and
p-completed
prime.
If the fixed point sets
X
G - CW
complexes
X , p
a
have all components nilpotent and of
finite type, then the canonical map H (XH; SZ/p) +HL((X ) H ; 2Z /p) * * P Definition 3.1. is called equivariant at
p,
X •> X will induce isomorphims P for all H
A map
f : X -* Y
if
f : X -> Y P P P
between
G - CW
complexes
is homotopic to an equi-
variant map. Such a map
f : X •*• Y
which is equivariant at
p
will give
rise to an induced map
p The notation
f* P
G
p
G
p
is however somewhat misleading since the induced map
may depend on the choice of the equivariant representative of
f ; we P will assume in the sequel that such a choice has been made once and for
Glover & Mislin: Finite group actions all.
Note that
* ~ n K (X ; 7L /p 7L ) p Also, if
X
180
K (X ; 7L ) ^ lim K (X ; 2 / p 2 ) since the groups P P — -*• P * have a natural compact topology; similarly for K . G
is finite and nilpotent, then
it
*
/s
/\
K (X ; 7L ) ^ K (X) 0 ZZ . P P ~ P
Thus the ordinary Chern character gives rise in this case to an isomorphism o
/v
A
ev
K (X ; 7L ) ® CD ^ H P P ~"
A
(X ; Z ) <2> CD. P P
Similarly, the equivariant Chern character of Slominska [17] can easily be adapted to the
p-completed situation.
Let us write
R & (J G p
for
the coefficient system (in the sense of Bredon) given by (R S> $ ) (G/H) = R(H) ® $ , Q = 7L €> CJ) the field of G p P P P Then for
X
a
G - CW
complex with
G
p-adic numbers.
finite and each connected com-
TT
ponent of
X
a finite nilpotent complex, there is a natural isomorphism ch
G
° * >* : K (X ; E G p p
ev A ) <S> $ + H^ (X ; R^ 0 $ ) G p G p
and the canonical map H
G V( V *G * V - H^G ""^p' Vffi®(RH ® V
is injective. The following generalization of our Main Theorem then holds. Theorem 3.2. of the finite group
G.
Let
divide
: G -> GL
((C) be representations
If there exists a map f :
of degree prime to
p , p
V t ^ l ' •" ^ s . t ^
|G| such that
f
is equivariant at all primes which
|G|, then there is a one dimensional representation
with the property that
p
and
X® p
are Galois conjugate.
X
of
G
Glover & Mislin: Finite group actions Proof.
Write
divisors of
(Qi i for |G| G.
11$ , p
181
the product being taken over all prime
The argument then is completely analoguous to the proof
of the Main Theorem, replacing
$
by
$i i at all the obvious places.
The Theorem then admits the following converse. Theorem 3.3. tions such that
p
Suppose
^ X <S) p
p ,
p
: G -> GL
for suitable
(£E) are representa-
a G Gal((C/Q))
and
X : G -* (C .
Then there exists a map f
with
:
^s t ( p l ) * ^
deg f = deg(a) mod |G| such that
which divide
f
t
( p
2}
is equivariant at all primes
|G|. TT
Proof.
The fixed point sets
£C s,t
for a linear action on
I
s,t
are
disjoint unions of products of complex Grassmann manifolds, which are nilpotent and finite.
The canonical map
which one obtains by realizing the CC
(p )
p-adic etale homotopy type of
(cf. Artin-Mazur [1]) is therefore an equivariant homotopy
equivalence (it induces a homology equivalence with
2Z /p
coefficients
on all fixed point sets). Since
p
^ p
for every
T G Gal ((C/g)) with
deg(i) E deg(a) mod |G|, we may assume without loss of generality that there exists a large prime number
N
deg(a) = N mod \G\
Clearly,
map
for
all
j.
(larger than a
(s + t)!)
such that
induces an equivariant
Glover & Mislin: Finite group actions and
deg a
6 7L *
is
N
for all primes
Friedlander [9] we can, since
p
182
which divide
(N, (s + t)i) = 1 ,
|G|. By
construct a global
map
with
g
%> a
for all primes
p
which divide
|G|. The map
by definition equivariant at all primes which divide p
'u X & p ,
|G|.
g
is then
Since
there is an equivariant homeomorphism
h : C (p ) -*• tC (p ) S/t 1n s,t 2n desired map with
of degree
1.
The map
f = h o g
is then the
deg(f) = deg(a) mod |G|.
Remark.
Instead of considering actions on Grassmann manifolds,
one could in a similar way study actions on generalized flag manifolds and one would obtain similar results, as long as the rigidity result of Ewing-Liulevicius still holds (cf. [8]). In particular, one can formulate and prove our Main Theorem for actions on the standard flag manifold U(n)/U(l) n .
Glover & Mislin: Finite group actions
183
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