Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
895 Jonathan A. Hillman
Alexander Ideals of Links
Springer-Verlag Berlin Heidelberg New York 1981
Author
Jonathan A. Hillman Department of Mathematics, University of Texas Austin, TX 78712, USA
AMS Subject Classifications (1980): 13 C 99, 57 M 25, 5 7 Q 45 ISBN 3-540-11168-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-11168-9 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to '.'VerwertungsgesellschaftWort", Munich. 9 by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
The characteristic polynomial of a linear map is one of the most basic of mathematical objects, and under the guise of the Alexander polynomial has been much studied by knot theorists.
The rational
homology of the infinite cyclic cover of a knot complement is indeed determined by a family of such polynomials.
The finer structure of the
integral homology, or the homology of covers of a link complement (corresponding to a set of commuting linear maps) is reflected in the Alexander ideals.
These notes are intended to survey what is presently
known about the Alexander ideals of classical links, and where possible to give "coordinate free" arguments, avoiding explicit presentations and using only the general machinery of co~autative and homological algebra. This has been done to clarify the concepts;
in computing examples it is
convenient to use Wirtinger presentations and the free differential calculus, Seifert surfaces or surgery descriptions of links.
(The
avoidance of techniques peculiar to the fundamental group or to 3-dimensional topology means also that these arguments may apply to links in higher dimensions, but little is said on this topic after Chapter II.) This work grew out of part of my 1978 A.N.U.
Ph.D. thesis.
However although most of the proofs are mine, a number of the results (mostly in Chapters I, IV, VII and VIII) are due to others.
Some of the
latter results have been quoted without proof, as the only proofs known to me are very different in character from the rest of these notes.
fV
I would like to acknowledge the support of a Co~mlonwealth Postgraduate Research award at the Australian National University while writing my thesis, and of a Science Research Council grant at the University of Durham while preparing these notes.
I would also like
to thank Professors Levine, Murasugi, Sato and Traldi for sending some of their (as yet) unpublished notes to me. thank Mrs. J. Gibson and Mrs. S. Nesbitt have prepared the typescript.
Finally I would like to for the care with which they
CONTENTS
PRELIMINARIES
CHAPTER I
LINKS AND LINK GROUPS
CHAPTER II
RIBBON LINKS
16
CHAPTER III
DETERMINANTAL INVARIANTS OF MODULES
27
CHAPTER IV
THE CROWELL EXACT SEQUENCE
4O
CHAPTER V
THE VANISHING OF ALEXANDER IDEALS
54
CHAPTER VI
LONGITUDES AND PRINCIPALITY
66
CHAPTER VII
SUBLINKS
82
CHAPTER VIII
REDUCED ALEXANDER IDEALS
lOO
CHAPTER IX
LOCALIZING THE BLANCHFIELD PAIRING
116
CHAPTER X
NONORIENTABLE SPANNING SURFACES
140
REFERENCES
153
INDEX
175
PRELIMINARIES
In these notes we shall generally follow the usage of Bourbaki [ |3 for commutative algebra, Crowell and Fox [ 43 ] for combinatorial group theory, and Rourke and Sanderson [159 ~ for geometric topology.
The
book of Magnus, Karrass and Solitar [ ]23 ] is a more comprehensive reference for combinatorial group theory, while the books of Hempel [157]
[ 69 ], Rolfsen
and Spanier [ ]77 ] are useful for other aspects of topology.
All manifolds and maps between them shall be assumed PL unless otherwise stated.
The expression A ~ B means that the objects A and B
are isomorphic in some category appropriate to the context.
When there is
a canonical isomorphism, or after a particular isomorphism has been chosen, we shall write A = B.
(For instance the fundamental group of a circle is
isomorphic to the additive group of the integers ~ ,
but there are two
possible isomorphisms, and choosing one corresponds to choosing an orientation for the circle).
Qualifications and subscripts shall often be omitted, when there is no risk of ambiguity.
In particular "~-component n-link" may be abbreviated
to "link", and the symbols A , X(L), G(L) may appear as A, X and G.
CHAPTER I
LINKS AND LINK GROUPS
This chapter is principally a resum~ of standard definitions theorems, without proofs.
Although our main concern is with the classical
case, we have framed our definitions We begin with definitions relations between them. links.
and
so as to apply also in higher dimensions.
of links and of the most important equivalence Next we consider link groups and homology boundary
There follows a section on the equlvariant homology of covering
spaces of link exteriors,
and we conclude with some comments on the construction
of such covering spaces.
Let ~ and n be positive integers.
If X is a topological
space, let
~X be the space X • {I, ...,~}, the disjoint union of ~ copies of X. D n = {<x , .... x > in ~ n n
Let
I I ~ Zi ~ n x.l 2 < I} be the n-disc and let
S n = ~D n+| be the n-sphere.
The standard orientation of ~ n [ 159
induces an orientation of D n, and hence of Sn-1 by the convention
; page 44] that the
boundary of an oriented manifold be oriented compatibly with taking the inward normal last
Definition
(cf E 159 ; page 453).
A y-component
n-link is an embedding L:uS n ---+ S n+2.
component of L is the n-knot link type is an equivalence ambient
(l-component
The i th
n-link) L. = L I Sn x {i}. i
A
class of links under the relation of being
isotopic.
Notice that with this definition,
and with the above conventions
the orientation of the spheres, all links are oriented. locally flat (essentially because S ~ in $2), but embeddings 2 need not be locally flat
A I-link is
there are no knotted embeddings
of higher dimensional manifolds E 161 ; page 5 ~ .
on
of
in codimension
Definition
An 1-equivalence
between two embeddings
F:A x [0,I] ---+ B x [O,I] such that F ] A x F
-I
f,g:A § B is an embedding
{O} = f, F [ A x
{I} = g and
(B • {O,l}) = A • {O,l}.
In this definition we do not assume that the data are PL (so here an embedding
is a l-I map inducing an homeomorphism
results of Giffen suggest that wild 1-equivalences context of PL links
A locally
[ 54a ].
with its image).
Recent
have a r61e even in the
Clearly isotopic embeddings are I-equivalent.
flat isotopy is an ambient isotopy
even an isotopy of l-links need not be locally flat.
[ 159 ; page 58], but For instance any knot
is isotopic to the unknot, but no such isotopy of a non trivial knot can be ambient.
However a theorem of Rolfsen shows that the situation for links
is no more complicated. Definition
Two ~-component
is an embedding j:D n+2
n-links L and L' are locally isotopic if there
~ Sn+2 such that D = L-I(j(Dn+2))
is an n-disc
in one component of ~S n and such that Ll~S n - D = L'[~S n - D.
Theorem
(Rolfsen
[ 153 ])
Tw__on-links L and L' are isotopic
if and only
if L' may be obtained from L by a finite sequence of local isotopies and an
ambient isotopy.
In other words L and L' are isotopic obtained from L by successively
if and only if L' may be
suppressing or inserting
small knots in
one component at a time.
Definition
A concordance between two ~-component
locally flat l-equivalence.~between (or slice) if it is concordant
L and L'.
n-links L and L' is a
A link is null concordant
to the trivial link.
A link L is a slice link if and only if it extends to a locally flat embedding C:~D n+l ---+ D n+3 such that C-|(S n+2) = ~S n.
Definition
Two ~-component n-links L and L' are link-homotopic if there
is a map H:~S n •
EO, I]----+ S n+2 such that HIuS n • {O} = L , H ] u S n • {I} = L'
and H(S n • {t} • {i}) 0 H(S n • {t} x {j}) = @ for all t in [O,1] and for all 1 .< i # j .< U.
In other words a link-homotopy is a homotopy of the maps L and L' such that at no time do the images of distinct components of uS n intersect (although self intersections of components are allowed).
Milnor [ 129]
has given a thorough investigation of homotopy of ]-links. Goldsmith
~ 57 ]
link-homotopic.
Giffen ~ 5 ]
and
have recently shown that concordant l-links are (Giffen
[54a]
links need not be PL I-equivalent).
has also shown that 1-equivalent For other results on isotopy of links
and related equivalence relations see ~26, 82, III, 130, 154, 155,175].
The link group
The basic algebraic invariant of a link is the fundamental group of its complement, and most of these notes are concerned with the structure of metabelian quotients of the groups of l-links.
Definition
The exterior of a U-component n-link L is X(L) = Sn+2 - N,
where N is an open regular neighbourhood of the image of L.
The group of
the link L is G(L), the fundamental group of X(L).
The exterior of L is a deformation retract of Sn+2 - L, the complement of L, and is a compact connected PL (n+2)-manifold with boundary components.
By Alexander duality HI(X(L);~) ~ ,
Hi(X(L))~)
= O
for ! < i < n+] and H n + ) ( X ( L ) ~ ) ~ all links are locally flat. for the i
th
~ ~-].
We shall assume henceforth that
Then ~X(L) = ~S n • S I.
A meridianal curve
component of L is an oriented curve in the boundary of X(L)
which bounds a disc in s n + 2 - - X ( L i ) having algebraic intersection +1 with L.. 1
The image of such a curve in the link group G is well defined up to
conjugation, and any element of G in this conjugacy class is called an .th i meridian.
The images of the meridians in the abelianization G/G' = HI(X:~)
are well defined and freely generate it, inducing an isomorphism with ~ .
An application of van Kampen's theorem shows that G is the normal closure of the set of its meridians.
(The normal closure of a subset S
of a group is the smallest normal subgroup containing S, and shall be denoted << S>>).
Thus the group of a ~-component n-link is a finitely
presentable group G which is normally generated by ~ elements, with abelianization ~ theorem H2(G;~)
and, if n ~ 2, with H2(G;~)
= 0
(since by Hopf's
is the cokernel of the Hurewlcz homomorphism
~2(X) ---+ H2(X;~) [ 83
~ ).
Conversely Kervaire has shown that if
n ~ 3 these four conditions characterize the group of a U-component n-link [ 96 ~.
If n = 2 these conditions are neccessary but not sufficient,
even for ~ = I [ 71
]; if the last condition is replaced by "the group
has a presentation of deficiency ~" then Kervaire showed also that it is the group of a link in some homotopy 4-sphere, but this stronger condition is not neccessary.
The case of 1-11nks with ~ > ] is quite different. H2(G;~)~
For then
~-l unless ~2(X) ~ O, in which case by the Sphere Theorem [ 147
the link is splittable.
(An n-link L is splittable if there is an (n+l)-
sphere Sn+l ~ Sn+2 - L such that L meets each complementary ball, that is, each component of Sn+2 - sn+l).
This is related to the presence of longitudes,
non trivial elements of the group commuting with meridians.
Let L be a ~-component ;-link.
An i th longitudinal curve for L is
a closed curve in the boundary of X(L) which is parallel to L. (and so in i .th particular intersects an i meridlanal curve in just one point), and which is null homologous in X(Li).
The i th meridian and i th longitude of
L, the images of such curves in G(L), are well defined up to simultaneous conjugation. If X(L) has been given a 5asepolnt *, then representatives of the conjugacy classes of the meridians and longitudes in ~I(X(L),*) ~ G(L) may be determined on choosing paths joining each component of the boundary to the base point.
The linking number s
13
of the i
th
component of L with
the j.th is the image of an ith longitude of L in HI(X(Lj);~) not hard to show that s
13
= s
31
(Notice that s
II
= ~
it is
= 0).
When chosen as above, the i th longitude and i th meridian commute, since they both come from the fundamental group of the i th boundary component, which is a torus.
In the case of higher dimensional links
there is no analogue of longitude in the link group, because spheres of dimension greater than or equal to 2 are simply connected, while in knot theory the longitudes are often overlooked, as for 1-knots they always lie in the second commutator subgroup G" (See below).
The presence of the
longitudes gives the study of classical links and their groups much of its special character.
If the i th longitude is equal to I in G(L), then L. extends to an 1 embedding of a disc disjoint from the other components of L, by the Loop Theorem [147 ].
A link is trivial if all the longitudes equal I.
Theorem 1
Proof of L).
A l-link L is trivial if and only if G(L) is free.
(Note that the rank of G(L) must equal the number of components Since a free group contains no noncyclic abelian subgroups [123; page 423 ,
the i th longitude and i th meridian must lie in a common cyclic group. considering the images in HI(X(Li);~ ) = ~ , we conclude that the i longitude must be null homotopic. we see that the link is trivial.
On
th
Hence using the Loop Theorem inductively The argument in the other direction is
immediate. H
This result may be restated as "An n-link is trivial if and only if
[n +l ],,
the homotopy groups ~j(X) are those of a trivial link for j ~ L-~-and in this form remains true for n-knots whenever n ~ 3. proofs are quite different.
See Levine
(Of course the
[ 114 ] for n ~ 4 and for n = 3
see Shaneson [ ]69 ~ in conjunction with Milnor duality [ 132 ]).
However
it is false for all ~ ~ 2 and n ~ 2, as was first shown by Poenaru [149 ]. (See also Sumners ~181 ] and the remarks following Theorem 11.6 below).
Definition
A ~-component n-link L is a boundary llnk if there is an
embedding P:W =U W. ---+ Sn+2 of ~ disjoint orientable (n+1)-manifolds i each with a single boundary component, such that L = PLOW.
All knots are boundary links, and conversely many arguments and results about knots proved by means of such "Seifert surfaces" carry over readily to arbitrary boundary links.
Theorem
(Smythe [ 174 ])
A B-component 1-1ink is a boundary link if
and only if there is a map of G(L) onto F(p), the free group of rank ~, carrying some set of meridians to a basis of F(~).
Guti~rrez extended Smythe's theorem to n-links and characterized the trivial n-links for n ~ 4 as the boundary links whose complement has
Ln+l~
the correct homology groups ~j(X) for j ~ L-~--J [ 61
].
The splitting
theorem of Cappell shows that this is also the correct criterion for n = 3 [ 22 ].
(Little is known about the case n = 2, even for knots.
See Swarup ~187 2).
Definition
A b-component link L is an homology boundary link if there
is an epimorphism G(L) ----+F(~).
Note that there is no assumption on the meridians. such an epimorphism is necessarily G
=
n~0 Gn'
The kernel of
the intersection of the
intersection of the terms of the lower central series of G. in this section).
(See below
Smythe showed also that a l-link L is an homology
boundary link if and only if there are ~ disjoint orientable surfaces U. in X(L) with ~U. C ~X(L) and such that ~U. is homologous to the i l l i
th
longitude in 3X(L).
(Such surfaces shall be referred to as "singular
Seifert surfaces").
For an homology boundary link, the longitudes lie in
Gm, since a free group contains no noncyclic abelian subgroups.
For a
boundary link, they lie in (Gw)' , since they bound surfaces which lift to the maximal free cover of the link complement. section).
(See also the next
Any l-link is ambient isotopic to a link L with image lying strictly above the hyperplane ~ 2
• 0 in ~ 3
poL with the projection p:]R 3 ___+ ~ 2 many double points.
= $3-{~} and for which the composition is local embedding with finitely
Given such a link, a presentation for the link group
(the Wirtinger presentation) may be found in the following way.
For each
component of the link minus the lower member of each double paint pair assign a generator.
(This will correspond to a loop coming in on a
straight line from =, going once around this component, and returning to ~). For the double point corresponding to the arc x crossing over the point -! separating arcs y and z, there is a relation xyx = z , where the arcs are oriented as in Figure 1.
Figure ]
This gives a presentation of deficiency 0 for G(L), of the form -I X.~ {x..,lj 1 ~ j ~ j(i), I ~ i ~ l u i j lJ uij = xij+l, I ~ j ~ j(i), l < i} • (where uij = Xpq for some p, q and xij(i)+ 1 = xi|). It is not hard to show that one of these relations is redundant.
72-86]
for details).
Thus a l - l i n k
(See Crowell and Fox ~3; pages
group has a presentation
For a knot group this is clearly best possible 9
of deficiency
1.
10
Theorem 2
The group G of a link L has a presentation of deficiency greater
than | if and only if L is splittable.
Proof
If G has a presentation with a generators and b relations,
then G
is the fundamental group of a 2-dimensional
cell complex Z with I O-cell,
a l-cells, and b 2-cells, so rank H2(G ;~)
~ rank H2(Z;~)
Therefore if a - b > I then rank H2(G;~) ~
~ - 2 < rank H2(X(L);~)
= rank Hl(Z;~)+b-a.
so ~2(X(L)) # O, and so by the Sphere Theorem X(L) contains an embedded essential 2-sphere which must split L. is immediate,
The argument in the other direction
since the group of a splittable link is the free product of
2 link groups. H
As in [ 62 ~ the group G(L) can be given a "preabelian" presentation [ 123 ; page 149] of the form {xi' Yij' 2 ~ j ~j(i),
1 ~i~l[vij
, xi~Yij , [wi, xi] , 2 ~ j ~j(i),
1 ~i~}
where the vi:3 and w.1 are words in the generators x~l and yi~j and where the .th word w. represents an i longitude in G(L). l
(Notice that the generator
x.I here, and all the generators x.. lj for I ~ j ~ j(i) in the Wirtinger presentation are representatives
Theorem
(Milnor [ ]30 J)
o f the i t h m e r i d i a n s ) .
The nilpotent quotient G/G
n
of a link group
G has a presentation of the form {xi, I ~ i ~ ~ I [wi(n), xi], l ~ i ~ ~, F(~) n }
where .... w.l(n) is a word in
the g e n e r a t o r s r e p r e s e n t i n g t h e image o f t h e i t h l o n g i t u d e .
These nilpotent quotients are of particular interest because of the following result.
11
Theorem
(Stallings [ 178 ])
If f:H ~
K is an homomorphism inducing an
isomorphism on first homology (abelianization) and an epimorphism on second homology (with coefficients in the trivial module ~) then f induces isomorphisms on all the nilpotent quotients f :H/H -~-+ K/K . n n n Consequently, if ~
is an 1-equivalence of two links L
and LI, then the O
natural maps G(Lo)/G(Lo)n-~+G(o~)/Gr
_ _
are isomorphisms, and so the
nilpotent quotients of the link group are invariant under 1-equivalence.
Here G ( ~
denotes ~I(S 3 x EO,I] -2f.).
If L is an homology boundary
link, the epimorphism G(L) ---+ F(~) satisfies the hypotheses of theorem, and so G/G
~
F(~)/F(~)
n
G/G [145
~+ F(~)/F(~)
for all integers n ~ I.
Stallings'
Hence
n
= F(~), since free groups are residually nilpotent
; page !12 ].
If G is the group of a higher dimensional link then
the inclusion of a set of meridians induces a map F(B) ----+ G which also satisfies the hypotheses of Stallings' theorem, so again G/G ~ F ( ~ ) / F ( ~ ) n
In this case however we cannot assume that the map F(~) ---+ G/G
. n
is onto,
although it is I-I.
Equivariant (co)homology
Let L be a ~-component n-link and let p:X' § X be the maximal abelian cover of the exterior of L.
On choosing fixed lifts of the cells of
X to X' we obtain a finite free basis for C,, the cellular chain complex of X', as a ~ ~/G']-module. enables us to identify = ~ / G ' ]
The isomorphism determined by the meridians with A~
=
~[~]
= ~ ~ l , t ~ 1,..
the ring of integral Laurent polynomials in ~ variables.
.t ,t ~- I ],
This ring is a
regular noetherian domain of dimension ~ + I, and in particular is factorial. As a group ring A~ has a natural involution, denoted by an overbar, sending each
t i to t. = t~ I, and augmentation g:A --+ ~ , i
I
which sends each t. to I. 1
12
Let Hp(X;A) denote the A-module Hp(C,), which is just Hp(X';Zg) considered as a
A-module via the covering transformations, and let
HP(x;A) denote the pth cohomology module of the dual complex HomA(C,,A). (This may be regarded as the pth cohomology with compact supports of X'). Since C, is a finite free complex and A is noetherian, all these homology and cohomology modules are finitely generated.
The cohomology modules may
be related to the homology modules by the Universal Coefficient spectral sequence [158
; page 347 ] : Ext~(Hp(X;A),A) ==> HP+q(x;A)
There is also a Cartan-Leray spectral sequence E158 Tor~ (Hq (X;A) ,~)
; page 345 ] :
Hp+q(X ;~)
relating the equivariant homology to the homology of the base. is a A-module via the augmentation map.
(Note that
We shall not need the corresponding
Cartan-Leray spectral sequence for cohomology).
If q:(~,~) ---+ (Y,Z)
is any regular cover of a simplicial pair, there are similar equivariant (co)homology modules and spectral sequences.
Now since X is a compact PL (n+2)-manifold with boundary, there are Poincar~ duality isomorphisms [ 131 ]: HP(x;A) ~-~ Hn+2_p(X,~X;A ) given by cap product with the orientation class in Hn+2(X,~X;~) .
(Here
if A is a A-module, A denotes the conjugate A-module, with the same underlying abelian group but with a in A, % in A).
A-action given by <%,a> ~--+ ~.a for all
This map may be interpreted geometrically in terms of
intersections of dual cells in X' as was done by Blanchfield E II ]. (See Chapter IX).
13
Other covers of a link exterior may Be treated in the same way. In particular if L is an homology boundary link we may consider the maximal free cover X m § X. ~/G
] ~ ~(~)]
In this case the coefficient ring
is coherent [200 ], so all the equivariant (co)homology
modules are finitely presentable, and of global dimension 2 , so the spectral sequences are fairly tractable.
However since this group
ring is not commutative, the distinction between left and right modules must be observed.
(Taking the dual or the conjugate converts left to
right and vice versa). Sato [162, 164] .
These facts have been applied to boundary links by
If L is an homology boundary 2-1ink, the Universal
Coefficient spectral sequence together with Poincar~ duality gives an isomorphism e2(G~/G~)
~
e 2 e2(G~/G~)
(where eq(M) = Ext~[G/G ] (M, ~[G/Gw])).
This isomorphism can probably
be used to show that there is a 3-1ink group which is not a 2-1ink group, although the groups of the component knots are 2-knot groups.
(In the
knot theoretic case such an isomorphism was used by Levine [ 120 ] to deduce that the p-local Alexander invariants of a 2-knot are symmetric and hence that not every high dimensional knot group
is a 2-knot group).
Covering spaces for link exteriors may often be constructed by splitting along Seifert surfaces.
This technique in conjunction with the
Mayer-Vietoris sequence leads to presentations of the equlvariant homology modules.
In the case of the first homology these presentations are often
more efficient than the Jacobian presentation, in that fewer generators or relations are needed.
This method has been used to construct infinite
cyclic covers of any link [ ]37 ], and the maximal abelian and maximal free covers of boundary links [63 ].
The latter construction works
equally well for homology boundary links, using "singular Seifert
14
surfaces" ~ 73 ] .
Recently Cooper has shown that the maximal abelian
cover of any l-link can be constructed in a similar way, on using Seifert surfaces which intersect
in a controlled manner E 34
also implicit in Bailey's thesis [
~.
(This idea is
7 ~).
We shall sketch the construction of the maximal free cover of an homology boundary link.
A map f:G(L) + F(~) corresponds to a map
F:X(L) + ~S I = K(F(~),I), which may be assumed transverse to {PI,...,P }, w h e r e P. i s a p o i n t 1 -1
Let W i = F
o f W = UW..
in the i th copy of S1 distinct
(Pi) and let Y = X-M There are
f r o m t h e wedge p o i n t .
where M is an open regular neighbourhood
two e m b e d d i n g s i + and i
1
o f W i n ~Y a nd --
X60 = Y x F(~)/
~ 3
< i (w.),h ~ for all w. in W., and h in F(~). -~3. 3 J
Here x. is a generator of F (~) corresponding to a loop in X which meets J W~ transversally in one point and avoids the other components of W. J
(This
is just the pull back of the corresponding construction of the universal cover of ~SI).
There is then a Mayer-Vietoris sequence:
.. § ZZ~F(~)~ ~)Hq(W) where dq(y ~ v j )
dq
ZZ[F(~)~ ~ H
q
(Y) .
= y xj O (ij+),(vj) - T ~
and v. in ~q(Wj;2Z). 3
. Hq(X~;ZZ . . .)
(i~),(vj)
for ~ in ~ F ( ~ ) J
If L is a boundary link we may assume that W. are J
Seifert surfaces; if L is also a l-link then since the longitudes bound the W., which llft to X60, they must be null homologous there, and so 3 lie in (G60)'. The Mayer-Vietoris sequence shows that for a boundary n-link Hn+I(X60;ZZ)
is a free 2Z~(~)~-module
of rank ~ - I.
If L is a boundary
1-1ink then H I (X60;ZZ) = G /G ' has a presentation with equal numbers of 6d
generators and of relations.
6O
(This remains true if L is an homology
T boundary link whose longitudes lie in G60).
15
A similar construction of the maximal abelian cover of the exterior of a boundary n-link together with an interpretation of the maps d
q
in terms
of Alexander duality in Sn+2 (as given by Levine for knots L If6 7) leads to the conclusion that these maps are monomorphisms if q ~ 1 and hence that p.d.Hq(X;A) ~ 2 for all q ~ 2. free of rank ~ - I.
For boundary l-links H2(X;A) is
(Sato has also investigated the homology of the
maximal abelian cover of the exterior of a boundary n-link [ 162 , 163 ]).
CHAPTER II In this chapter we introduce most of our examples,
the class of links that shall provide
and we establish
verification
of these examples.
Definition
A ~-component
R : ~D n+! § S n+2
RIBBON LINKS
several properties useful in the
n-ribbon map is an in~nersion
with no triple points and such that the components
the singular set are n-discs whose boundary ~S n = ~(~D n+l) ~-component
(local embedding)
("throughcut")
(n-l)-spheres
of
either lie on
or are disjoint from ~S n ("slit").
A
n-link is a ribbon link if there is an n-ribbon map R such
that L = RI~(~Dn+I). A ribbon l-link may be depicted
schematically
I16
Figure
l
as in Figure I .
17
It is easy to see that if L is a ribbon link, the ribbon R may be deformed
so that each component of the complement
bounded by at most two throughcuts.
of the throughcuts
is
In what follows R will always be
so chosen. It is well known and easy to see that ribbon links are null concordant
E49 ].
even for knots [81,
206
Theorem
[ 50 ; Problem
I.
Let L be a ~-component
ribbon n-link.
Then L is a sublink
ribbon n-link L for which surgery on the longitudes
~'~(S 1 x sn+l).
Proof.
25 ], but is false in higher dimensions
] as will be shown below.
of a v-component gives
The converse remains an open conjecture when n = |,
In particular
Let R : ~D n+! * Sn§
L is an homology boundary
be a ribbon extending L.
link.
Let S i ,
! ~ i ~ o, be the slits of R and for each slit choose a regular neighbourhood
N i contained
and such that N i ~
in the interior of the corresponding
Nj = ~ for i # j.
disc
Let v = ~ + ~ and let
= RI(~SnLj
U aNi). Clearly L is a v-component ribbon n-link with 1~i~ o L as a sublink. If n > I the normal bundle of L in S n+2 has an essentially unique framing;
if n = l give each component of L the 0-framing.
Let
W(L) = D n+3 U vD n+I x D 2 where T : vS n x D 2 § S n+2 is an embedding of a T regular neighbourhood of L = TIvs n • {0} determined by this framing. Then aW(L) is the result of surgery on Sn+2
along the longitudes
of L.
Now by adding a pushoff of LIaN i to the component of L bounding
the
(n+l)-disc containing Ni, L may be replaced by a ribbon link with one less singularity; the 0-framing.
moreover
if n = l each component of the new link still has
Continuing
which the only singularities
thus L may be replaced by a ribbon link L for are those corresponding
to the components
aN.. 1
Clearly these components may be slipped off the ends of the other
18
components
of the new ribbon and so ~ is a trivial ~-eomponent
adding pushoffs [105])
of link components
corresponds
to sliding
which leaves unchanged
to one another
(n+ l)-handles
the topological
Thus ~W(L) is homeomorphic
(a Kirby move of type 2
of W(L) across one another,
1 • Sn+;) and G(L) maps onto
If n = I, the kernel of the map G(L) ~ F(~) is necessarily is trivial,
link,
to a basis.
theorem
for 2 ~ i < n.
E61
This is so if and only if L is a boundary
], since ~i(S n+2 - imL)
if n > 2, by Guti~rrez' ~
~i(S n+2 - i m P )
The above theorem is not the best possible,
fewer new components may suffice to trivialize L thus. L is the square knot,
and
but need not carry any set
in which case L (and hence L) is trivial,
unlinking
G(L)m,
in which case L is also trivial.
If n > 1 the map G(L) § F(~) is an isomorphism, of ~ meridians
Now
type of W(L) and hence of ~W(L).
to ~W(~) = ~ ( S
is trivial if and only if L
link.
= 0
in that
For instance,
if
(figure 1 shows that) it is a component of a
2-component homology boundary
link with the above property.
Recalling
that a l-knot is said to have Property R if surgery on a longitude of the knot does not give S I • S 2, and that it has been conjectured non trivial l-knots have Property R ~O6;
Problem
that all
1.16] , this example
shows that the most direct analogue of Property R for links fails already for a 2-component for which 0-framed
link.
Is there a non trivial boundary
surgery gives a connected
l-link
sum of copies of S I • S 2 ?
19
Kirby and Melvin showed that any knot which does not have Property R is (TOP) null concordant
~|07 ], and this suggests the following complement
to the above result. Theorem 2
If n ~ 2 and L is a v-component n-link such that surgery on
the l o n g i t u d e s of L g i v e s - r 1 6 2 1 x s n + l ) , link and is null concordant.
then L i s an homology boundary
(Hence also any sublink of L is null
concordant).
Proof.
That L is an homology boundary link is clear.
the trace of the surgeries on L, so ~U(L) = sn+2_U_ D(L) = U(L) U ~ ( D 2 x sn+l)
Let U(L) be
~ ( S I x sn+l).
Then
is a contractible (n+3)-manifold with
boundary Sn+2, and so is an (n+3)-disc.
The link L clearly bounds v
disjoint (n+1)-discs in D(L). //
If n = I it can be proven that L bounds v embedded discs in a contractible 4-manifold Wo, by imitating the first part of the theorem of Kirby and Melvin. W o is D
Whether the Mazur trick may be used to show that
may be related to the Andrews-Curtis conjecture El06 ; Problem 5.2].
This comment is due to Rubinstein, who has also recently proven that if 0-frsmed surgery on the longitudes of the first p components of L gives P ~-(S I x $2), for each O,
20
Theorem 3.
A finitely presentable group G is the group of a V-component
sublink of an n-link L : v Sn + Sn+2 with group free (for some v and any n ~ 2), if and only if G has a presentation of deficiency V and is normally generated by V elements.
(If n = 2 the ambient space may be merely a
ho~otopy 4-sphere!) Proof.
The necessity of the condition is obvious.
presentation {Xi, ! $ i ~ vlr j ! ~ j ~ v - ~
Suppose that G has a
and that S k, I $ k ~ V, are
words in the symbols X i whose images generate G normally. fundamental group of
~ ( S 1 x Sn+]) is isomorphic to F(v), and the words I x sn+l)
rj and ~ may be represented by embeddings pj : S I § ~k : S1 § pj,
v
(S1 x
I ~ j ~ v-V,
sn+l )
1 x
respectively.
and ~ k '
homotopy (n+2)-sphere, v ~(S
The
sn+l )
If
surgery
is performed
I g k ~ V, t h e n t h e r e s u l t i n g
manifold
and
on a l l is
the
a
and
-
v-V V l U pj(S I x D n+l) - UOk(Sl x D n+ )
is the complement of a v-component n-link in this homotopy sphere with fundamental group F(v)
[96] o
Therefore if surgery is performed only on
the p j, I ~ j $ ~-V, the space v ((4#(S 1 x Sn+l) - v-V U oj(S I x D n + l ) ) ~ is
v-V U (D 2 x sn)) -
the complement of a V-component sublink
{Xi,
1 ~ i g v[rj,
1 g i g v-V},
that
is,
U~k(Sl x Dn + l )
with group presented
by
w i t h g r o u p G.
If for instance G.. lj is the group with presentation {XI,X2,X31XII
[x3i,xI ] ExJ,x2 ~ }
where ij ~ 0, then according to
Baumslag [;0~, Gij is parafree but not free, and so cannot map onto F(2). Thus the link constructed as above from this presentation is not an
21
homology boundary link, although it is a sublink of a 3-component homology boundary link.
(Notice that G.. is the normal closure of the ij
images of X 2 and X3, and the presentation i {XI,X2,X 3]x~ I [ X 3,El ] E X ~ ,X 2 ],X2,X 3 } of the trivial group is AC-equivalent to the trivial presentation, so this group can be realized by a 2-1ink in S 4 [120]).
We shall show in Chapter •
that a ribbon l-link need not be
an homology boundary link, although by Theorem 3 it is a sublink of one. In contrast a sublink of a boundary link is always a boundary link. An immediate consequence of Theorems I and 3 is that if n > I the group of a W-component ribbon n-link has a presentation of deficiency ~.
There-
fore for instance Fox's 2-knot with non principal first Alexander ideal E 49 ] is slice [ 96 ] but not ribbon.
(This was shown earlier by Mitt [81]
and Yanagawa [ 206] ).
Let R be a ribbon map extending the l-link L.
We shall show that
although the group of an unsplittable l-link has no presentation of deficiency greater than I, in the case of a ribbon l-link such a ribbon R determines a quotient of the link group which has deficiency ~.
Each
throughcut T of R determines a conjugacy class g(T) C G(L) represented by the image of the oriented boundary of a small disc neighbourhood in R of the corresponding sllt.
(The standard orientation of D 2 induces an
orientation on this neighbourhood via the local homeomorphism R.) Definition
The ribbon group of R is the group H(R)
=
G(L) / << Ug(T)[T a throughcut of R > > .
(Recall that << S >> denotes the smallest normal subgroup of G containing S.)
22
Lemma4
The longitudes of L are in << Ng(T) IT a throughcut of R > > .
Proof
Each longitude is represented (up to conjugacy) by a curve on and
near the boundary of the corresponding disc, which is clearly homotopic to a product of (conjugates of) loops about the slits in that disc. (See Figure 2 ) . H
Figure 2 Lemma5 Proof
For all throughcuts T, g(T) C G . Certainly, for all T, g(T) C G I.
their images g(T) are central in G/Gn+ 1 .
Suppose all g(T) C G n.
Then
It then follows that
g(T) = g(T') where T' is either throughcut adjacent to T, and hence, moving along the ribbon, that all g(T) = {l}, in other words that all g(T) C Gn+ 1 .
(See Figure
9.
By induction, all g(T) C G
9 .
Figure 3
j(T)
-
~cr')
.H
,-.,,,,/3c s )
23
Theorem 6
For any ribbon link L the projection G § G/G
factors as
G § G / << longitudes >> § H(R) § G/G , and H(R) has a presentation of deficiency ~ of the form {xij , 1 ~ j ~
j(i), 1 ~ i ~ l w i j x i j w l j
-i
= xij+l , 1 ~ j < j ( i ) ,
1 ~i~}
where there is one generator x.~ for each component of the complement i] of the throughcuts and one word w.. of length one for each throughcut. lJ (Here R is any ribbon map extending L which satisfies the condition imposed after Figure I). Proof
The factorization of G § G/G
follows from the lemmas.
It may
be assumed that in a generic projection of the ribbon there are no triple points.
The Wirtinger generators of the link group corresponding
to the subarcs of the projection of the link which "lie under" a segment of the ribbon may be deleted, and the two associated relations replaced by one stating that either adjacent generator is conjugate to the other by a loop around the overlying segment.
Figure 4
(See Figure 4).
24
Any loop about a segment of the ribbon is killed in H(R), for the only obstructions to deforming it onto a loop around the throughcut at an end of the segment are elements in the conjugacy classes of the throughcuts between the loop and that end.
(See Figure 5).
Figure 5
Hence the remaining generators corresponding to subarcs of the boundary of a given component of the complement of. the throughcuts coalesce in H(R).
Conversely the presentation obtained from the Wirtinger
presentation by making such deletions and identifications has the enunciated form, and presents a group in which the image of each g(T) is trivial, for the image of g(T) is trivial if and only if the pair of generators corresponding to arcs meeting the projection of T are identified.
Thus the group is exactly H(R).
H
25
Conversely R : ~D 2 + S 3.
ribbon n-link for n ~ 2 if and only if G has a Wirtinger of deficiency
to meridianal
to the components
correspond
of the complements
~=I
by Yajima
E205]~
See
Simon
of
(This
[173] and
for the connection between abstract Wirtinger presentations
and homology.) such
The generators
and there is one relation for each throughcut.
was proven for n = 2 , D86~
~ and G/G' = ~ .
loops transverse
the throughcuts,
Suzuki
can be realized by some ribbon map
A similar argument shows that a group G is the group of
a ~-component presentation
any such presentation
In Chapter VI we show that Baomslag's
group
G_I,I has
a presentation. Most of Theorem 6 can be deduced easily from Theorem i, by arguing
as in T h e o r e m 3
to adjoin ~ - ~
relations
G / < < longitudes >> , H(R) and G/G (For example,
consider
with two throughcuts.) knotting
to F(~) ~
are distinct groups,
be characterized
die.
as
Can H(R)
link- or group-theoretically?
2-1ink of which L is a slice,
H(R) is
and where the longitudes
of a clearly
doubling this nullconcordance
2C : ~S 2 + S 4 of which L is a slice,
this 2-1ink may be computed
the group
of the ribbon apart in D 4, a null
C : ~D 2 + D 4 of L is obtained;
gives an embedding
particular
then H(R I) = H(R2),
do not change the pattern of the singularities.
(By pushing the singularities
concordance
even when ~ = I.
If one ribbon R 1 is obtained from another R 2 by
As a partial answer to the above question, ~-component
In general
the square knot as the boundary of a ribbon link
the ribbon or by inserting full twists,
such operations
G/G .
to be H(R) by Fox's method
and the group of [48
the link of Figure ! is a slice of a 2-component
].) 2-1ink
Thus in
26
with group free.
Since this I-link is not a boundary link (see
Chapter VI) the 2-1ink is nontrivial and so this gives a simple example of the phenomenon first observed by Poenaru [149].
(This example was
given by Sumners, from a different viewpoint.)
Similarly the knot in
Figure 6 is a slice of a 2-knot with group ~.
Yanagawa
[207] has
shown that it is in fact a nontrivial slice of a trivial 2-knot.
1 Figure 6
He showed there also that a ribbon 2-knot with group Eg is trivial. In [193] Tristram shows that concordance of B-component l-links is generated as an equivalence relation by concordances of the form L ---+ L +b DR, where R : ~D 2 ----+ S 3 - imL is a ribbon map with image disjoint from that of L, and where +b denotes (iterated) band connected sum.
CHAPTER III Throughout
DETERMINANTAL
INVARIANTS
OF
MODULES
this chapter R shall denote an integrally
noetherian domain.
closed
(Although it would suffice for our applications
in the next section to assume further that R be factorial, not alter the proofs.
In fact as our principal
this would
technique is to
reduce to the case of a discrete valuation ring by localizing
at
height one prime ideals, most of our results may be extended to the case of an arbitrary Krull domain.)
Let M be a finitely generated R-module. is the dimension of the vector space M o = R o ~ R fractions
of R.
The R-torsion
The rank of M over R M over Ro, the field of
submodule of M is tM = {m in M ] r.m = 0
for some nonzero r in R } , and M is an R-torsion module if tM = 0. annihilator
ideal of M is A n n M
= ~r in R [ r . m
The
= 0 for all m in M } .
Let RP Q be a finite presentation
Rq _ ~
for M.
M ___+ 0
This presentation has deficiency q - p ,
and is said to give a short free resolution of M if the map Q is injective.
For each k ~ 0 the k th elementary
Ek(M) generated by the (q-k)• representing
determinantal
[43 ; page
101] .
ideal by Bourbaki
ideal by Buchsbaum and on the presentation and Fox.)
subdeterminants
Q if k < q and by ! if k ~ q.
Crowell and Fox
Eisenbud
(We use the terminology of
[13 ; page 573 ] and the (k+l) st Fitting [19 ] .
is well known,
system S in R.
of the matrix
This ideal is called the k th
Clearly Ek(M) ~ Ek+I(M)
multiplicative
ideal of M is the ideal
That it depends only on M, not
and is proven for instance by Crowell and Ek(Ms) = Ek(M) S
for any
28
For each k ~ 0 let AkM be the k th exterior power of M El4 ; page 507 ]
and let ~k M = Ann AkM.
The notation ~kM is due to Auslander
and Buchsbaum E 5 ] who showed that if R is local and ~kM is principal for all k then M is a direct sum of cyclic modules, and used this to give criteria for projectivity.
Since Ak(M) S = (^kM)s ~13;
page 78]
it follows that ~k(Ms) = (~kM)s ,) while clearly ~kM ~ ~k+l M.
In the
next result, relating these ideals to the elementary ideals, we shall invoke Cramer's rule in the following form. and let d # 0 divide each of the (a-l)•
Let A be an axa R-matrix subdeterminants of A.
If
u is an axl column matrix (respectively, a Ixa row matrix) then ((det A)/d) u
is an R-linear combination of the columns (respectively,
rows) of A.
See ~14; page 535~ 9
Theorem I
Let M be a finitely presentable R-module.
(i)
Eo(M) ~ A n n M
(ii)
~k(M)
Proof
(i)
Then
= ~I(M);
= ~k+|(M)
for each k ~ 0.
We may clearly suppose Eo(M) ~ O.
Let D be a q x q sub-
matrix of a presentation matrix Q for M (as above), with ~ = d e t D zero.
Then by Cramer's rule ~ 9 R q ~ D(R q) ~ Q(R p)
6.~(u) = ~(~.u) E im ~.Q = 0 every generator of Eo(M) (ii)
Let ~
for all u E R q.
and thus assume that ~
and so
Hence ~ E A n n M
and so
is in A n n M .
be a prime ideal of Ro
if and only if ~k+l(M) ~
non-
.
We must show that Ek(M) ~
We may localize with respect to R - ~
is the unique maximal ideal of R.
the dimension of the vector space M / ~ M
Let q be
over the field R/~ .
Then
2g
~k(M/~M)
= 0 if k ~ q and eq+1 (M/~M) = R/~ , so Ok(M) ~
only if k ~ q.
if and
By Nakayama's le~m~a [ 4; page 21 2, M has a
presentation with q generators.
Since M / ~ M
has dimension q, all the
entries of the presentation matrix are in ~ , and hence Ek(M ) ~ and only if k < q, that is, if and only if ~k+|(M) ~ ~k(M)
= n{~
prime I Ek(M) C ~
} = N{~
These results are well known.
.
prime I ~k+l(M) ~
if
In other words } = ~k+|(M).
(See for instance [15 ; page 573] ).
In [ 19], Buchsbaum and Eisenbud show also that for each k ~ 0 Ek(M) ~ ~k+l(M) ~ (Ek(M): Ek+I(M))
and give sufficient conditions for
this inclusion to be an equality.
(Their methods apply to modules over
any commutative noetherian ring.) Definition [ 13; page 476]
The divisorial hull I of an ideal I of R is
the intersection of the principal ideals of R which contain I. It is clear that if S is a multiplicative system inR then (Is)~ = (1)S
as ideals of the localization RS, while if R is factorial
and I # O the ideal I is a principal ideal, generated by the highest common factor of the elements of I.
Lemma 2
The divisorial hull of an ideal I is N Ip ' __
the intersection of
all of its localizations at height one prime ideals p of R.
Proof
Since R is an integrally closed noetherian domain, it is a Krull
domain, so R = nRp is the intersection of all of its localizations at height one primes, and these are each discrete valuation rings [13 ; pp.480-485].
Therefore if I is an ideal, (Ip) ~ = Ip so I _C N(1)p =
~
n(Ip) (a),
= nip .
On the other hand if I -C (a) then NI p - - C n(a)p = (a).NR p
so N I p-C i.H
30
If R is factorial and M is a finitely generated R-module, let Ak(M) be any generator of the principal ideal Ek(M)~, for each k ~ 0.
Lemma 3
If R is a discrete valuation ring and M is a finitely
generated R-module of rank r, then ~k M = 0 if = (Ar+j-I (M)/Ar+j(M)) ~r+j M ffi ~.tM j Proof
k ~ r and
for each j ~ I.
Let p be the maximal ideal of R.
By the structure theorem for
finitely generated modules over principal ideal domains, M~
R r ~ tM ~ R r ~ (I ~ i~ n
(R/pC(i))) where 0 < e(i) ~ e(i+l) for 1 $ i ~ n.
Therefore Ek(M) = O if k < r and Er+j(M) = E.j(tM) = pSj where s- = J
~ 1 ~
AkM~
e(i) for each j >. O.
O.<j.<(~k(Aj (Rr) ~ ~ _ j t M )
[|4;pages 515-518] j ~ I.
Moreover
0<j.~ .
=
and hence ak M = O if k < r and ~ ~ " r+j
Therefore we may assume r = 0. AkM ~
~
< .. < i ( k )
1 .
< .. < i ( k )
~
~
Then
((R/p e(i(1))) ~ ) . .
1 .
= ~.tM for each j
@ (R/pe(i(k))))
.
(R/p e(i(1))) .
(R/pC (i)) f(i)
l.
where f(i) = Card{(i(2) .... i(k)) in ~ k - l l i
< i(2) < ..
Clearly f(i) = 0 if i > n-k+l and f(n-k+l) = I. ~k M = Ann(AkM) = pe(n-k+l)
Therefore
= (Ak_I(M)/Ak(M)). H
~ n}.
31
If we note that the above ideal (Ar+k_l(M)/&r+k(M)) quotient
is the ideal
(Er+k,l(M) : Er+k(M)) = {S in RIS.Er+k(M) ~ Er+k_l(M)} when R is
a discrete valuation ring, then we may extend this result as follows.
Theorem 4
If M is a finitely generated R-module,
then ~kM = 0 for each
k $ r = rank M and (~r+jM) ~ = (Er+j_I(M) :Er+j(M)) ~ for each j ~ I.
Proof
By Lemma 2 it is enough to prove that the localizations of the
ideals at height one primes are the same.
This follows from Lemma 3 on
observing that every step (forming exterior powers, annihilators
etc.)
is compatible with localization. //
Corollary (~r+jM)
If the coefficient ring is factorial, = (Ar+j_I(M)/Ar+j(M))
for each j ~ I.
then Hence (Ar+j(M)) = (Aj(tM))
for each j ~ 0. ~
Proof theorem.
Remarks
By Lemma 3 (~r+jM)
~
= (~jtM)
for each j ~ I.
Now apply the
//
I. The second part of this Corollary was proven by Blanchfield
[II ; Le~mla 4.10~ . 2. If M is a torsion module
(over a factorial domain) which has
a square presentation matrix then it follows easily from Cramer's rule that (Ao(M)/AI(M)) ~ Ann M. Ann M ffi (Ao(M)/AI(M)).
The Corollary then implies that
(Buchshaum and Eisenhud show that if R is any
noetherian ring and M is an R-module with a square presentation matrix whose determinant
is not a zero divisor, then Ann M = (Eo(M):EI(M))
[19]).
32
Le~ma 5
Let 0 § A § B ~ C § 0
be an exact sequence of R-modules
that A is an R-torsion module and rank B = rank C = r.
such
Then
Er(B) ~ = Eo(A)~Er(C) ~ .
Proof
If R is a principal
consequence
ideal domain,
of the structure
then follows on localization
this is an immediate
theorem for R-modules.
The general case
at height one primes.//
This lemma was first proven by Levine and A, B and C all R-torsion modules~
[I17~
for R factorial
If C has a short free resolution
we can do slightly better.
Lemma 6
Let 0 § K § M ~ Q § 0
Then Ei(M) ~ E o ( K ) E i _ j ( Q ) resolution
Proof
be a short exact sequence of R-modules.
for all i ~ j.
If Q has a short free
and is of rank r, then Er(M ) = Eo(K).Er(Q)-
Given presentation matrices
P(K) and P(Q) for K and Q
respectively
it is easy to see that M has a presentation matrix of the
form
] p0( Q ) .
[ P(~ )
generators
(Here we assume the columns correspond
and the rows to the relations.)
for all i ~ j.
to the
Hence El(M) ~ Ej(K)Ei_j(Q)
If Q has a short free resolution we may assume that
P(Q) is a q • ( q + r )
matrix, where r = rank Q.
It is then easy to see
that the only nonzero elements of Ei(M) are those obtained by deleting r columns form P(Q) and taking the product of the resulting Er(Q) with a subdeterminant
Definition
~ 13;
page 523]
of P(K) of
column index O.
element
//
An R-module M is pseudozero
if M
= 0 P
for every height one prime ideal p of R.
of
SS
Lemma 7
i) ii)
M is pseudozero if and only if (Ann M) ~= R. I f N is a pseudozero submodule of M, then (~kM) ~ = (~k(M/N))~
and Ek(M)~ = (Ek(M/N))~ for each k ~ O.
Proof
Since (~kM)p = ~k(Mp) and Ek(M) p = Ek(M p) for each k ~ O, the
first assertion follows from Lemma 2 on considering ~IM and the second follows from the equations Ek(M) p = Ek(M p) = Ek((M/N)p) Lemma 8
(~kM)p = ~k(Mp) = ~k((M/N)p)
= (~k(M/N))p and
= Ek(M/N) p. //
Let A be a bxc R-matrix of rank d, and suppose there is a
dxd submatrix D such that det D divides every dxd subdeterminant of A. r
Then there are invertible square matrices B and C such that BAC = [~ ~J
Proof
After permuting the rows and columns if necessary, we may
assume that D is in the top left hand corner of A.
We may then apply
Cramer's rule to annihilate the partial rows and columns below and to the right of D.
The bottom right hand corner block of the resulting
matrix must be null as rank BAC = rank D. //
Recall that a nonzero ideal I of a noetherian domain R is called invertible if it is projective as an R-module, and that a finitely generated module M is projective if and only if the localization M
is P
a free R -module for each prime ideal p of R [13;page 109]. P Theorem 9
Let M be a finitely generated R-module of rank r.
Then
Er(M) is invertible if and only if P = M/tM is projective and tM has pro~ective dimension at most one. Er+ j(M) = Ej(tM) for each j >, O.
In this case M ~ P
9 tM and
34
Proof
Suppose that E (M) is invertible, r Rp
Q~- R q
Let p be a prime ideal of R.
~ mM
and that M has a presentation
~-O.
On localizing
at p the hypothesis
le~m~a B is satisfied by the matrix Q, for any generating principal
set of a
ideal in a local domain must contain a generator,
Nakayama's
lemma .
Thus M
~ P
of
by
R r ~ tMp, where tM has a short P P
free resolution
0 ~
and P
R ~-r n P
= M /tM is free. P P
P
D-->Rq-r P
Therefore
=, tM
~ 0, P
P
is projective,
so the projection
P
of M onto P = M/tM splits, and if ~ : R c § tM is any epimorphism, ker ~ is locally free (e.g. by Schanuel's projective, then P
so p.d.tM ~ I.
is free and tM P
Conversely,
lemma
if P is projective
square matrix since tMp is a torsion module. Er(P p @ tM ) = Eo(tM) p p last assertion
Corollary
Proof
is principal
is clear.
[l~;Lemma
~
Hence Er(M) p = Er(M p) = The
//
M is projective
pseudozero
Let N he a pseudozero Then N
is a pseudozero P
if N
given by a
and so E (M) is invertible. r
If M is a finitely senerats
then M has no nontrivial
of R.
and p.d.tM $ I,
if and only if Er(M) = R.
This follows from the fact that tM = 0 if and only if Eo(tM) = R. //
Theorem 10
Proof
E|58;page 92~) and hence
has a short free R -resolution, P
P
then
R-module of projective
dimension
submodules.
submodule of M and let p be a prime ideal submodule of M . P
= 0 for all p, and since localization
Since N = O if and only
is exact, we may therefore
P assume that R is local, and hence that M has a short free resolution
O--->R p
Q > Rq --i-~ M ---~ O.
I,
35
We shall proceed by induction on r = q-p, the rank of M.
If r = O then E o(M) is principal and so Eo(M) N = Eo(M). N is p s e u d o z e r o ,
Since
E (M/N) ~ = E (M) ~ b y Lemma 7 a nd so t h e i n c l u s i o n s o o
Eo(M)~ Eo(M/N)~E
o
has a presentati~
(M/N) ~ are all equalities. matrix ~
the f~
I~ 8I which by 1 the argument ~
may be changed to [~] by row operations. onto M/N is an isomorphism,
Suppose r ~ I. submodule of M.
Since M/N is a quotient of M it Lemma
Hence the projection of M
so N = O.
Then we may assume that ~(eq) generates a free
(Here {el, ... eq} denotes the standard basis of Rq.)
Let M' = M/R.~(eq)
and let f:M + M' be the natural projection.
Then M'
has a short free resolution and its rank is r-l, so by the inductive hypothesis M' has no nontrivial pseudozero submodule. monomorphically
to M', as H.~(eq)
the theorem is proven.
Remark
is torsion free.
But f maps N
Therefore N = 0 and
//
If R is factorial and M is a torsion module then E (M) principal o
implies that M has no nontrivial pseudozero submodule
(by Theorems 9 and I0),
and since (Ann M ~ .M is certainly pseudozero this implies that Ann M is principal.
However in general these implications are strict.
For instance
if R = ~ Ex,y] then R/(x) @ R/(x,y) has annihilator ideal principal while its second summand is pseudozero,
and any maximal ideal of ~ [~considered
as an R-module in the obvious way has no nontrivial pseudozero submodule but its O th elementary ideal is not principal.
Going in the other direction we may characterize pseudozero modules homologically. The module e~
For each k ~ O, let ekM denote Ext~(M,R).
= HomR(M,R)
is torsion free, and there is a natural
36
"evaluation"
homomorphism
ev M :M § e~176
is a torsion module if and only if e~
with kernel tM.
= O.
Therefore M
If M is a torsion module then *
elM = HomR(M,Ro/R) 0 § R + R
(as follows on applying EXtR(M,-)
§ R /R § 0) and there is an "evaluation"
o
to exact sequence homomorphism
o
WM:M + elelM.
Theorem
II
Let M be a finitely senerated R-module.
is a torsion module with no nontrivial is
the
maximal pseudozero
pseudozero
submodule of M.
Then HomR(M,Ro/R) submodule,
ker WtM
Hence M is pseudozero
and only if e~
= elm = O.
Proof
.. mg generate M and let f:M + Ro/R.
Let ml,
and
If f(mi) = ri/s i mod R
(with s. # O~ then s.f = 0 with s = ~g s. # O, and so Hom~(M,Ro/R) i i=l l 9 torsion module.
If f
if
is a
= 0 for all height one prime ideals p then for each P
m in M f(m) is in the intersection
of all the localizations
R
and hence P
is in R, since R is a Krull domain. nontrivial structure
pseudozero
submodule.
+ eleltM P
pseudozero
It is an i~unediate consequence
of the
I
Theorem
11
Hence kerWtM is
assertions
follow readily. //
implies that if M is a finitely generated
supporting a nondegenerate
then M has no nontrivial Blanchfield
is an isomorphism.
ideal
P
and the remaining
torsion module
2
has no
theorem for finitely generated modules over a principal
domain that WtM,p:tM
Remarks
Therefore HomR(M,Ro/R)
pseudozero
bilinear pairing b:M • M § R /R o
submodule.
This was proven by
[11].
It is proven in [I13]
that if R is a regular local ring of
dimension d and M is a finitely generated R-module then M has no nontrivial submodule of finite length if and only if edM = O.
By localizing at
37
maximal
ideals,
it follows that this remains true for modules over a
noetherian domain all of whose localizations
at maximal
ideals are
regular of dimension d.
We conclude invariant.
this chapter by considering
another determinantal
Let M be an R-module with a finite presentation
Rp
Q* R q
and suppose that rank M = r. (q-r) th compound Q(q-r)
~ >M--+O
Then the matrix Q is of rank q-r and the
is of rank I.
Steinitz E179]
and Fox and Smythe [51]
showed that the ideal class of the ideal generated by the elements of any one column
(respectively,
row) of Q(q-r) depends only on the module M.
(Two nonzero ideals l and J belong to the same ideal class if there are nonzero elements
a
and
b
in R such that al = hJ.)
Let 7(M) and o(M)
denote the column and row ideal classes of M, respectively.
It is easy
to see that two nonzero ideals are in the same ideal class if and only if they are isomorphic
as R-modules,
and that every finitely generated
torsion free R-module of rank l is isomorphic in mind,
the Steinitz-Fox-Smythe
to an ideal.
With this
row invariant may be characterized
as
follows.
Theorem 12
The row ideal class p(M) is the isomorphismclass
rank ] torsion free module
Proof
of the
(~rM)/t(~rM).
Let U be a (q-r)Xq submatrix of maximal rank q-r of the presentation
matrix Q.
Define ~: (Rq) r § R by ~(V 1 ,.. .V 1 ,... v r) = d e t ~FVtr tr ''" .v~rut~ 1 '''" V r
38
where the vectors V I ' " ' V r q • q matrix.
in R q are used as the first r columns of a
The map # is clearly alternating (for if two of the
arguments V. are interchanged the sign of the determinant changes) and i ~(VI,...V r) = 0 if any of the arguments V. are in the image of Q, so i factors through A M. r ~:(ArM)/t(~M)
Since R is torsion free there results a map
+ R which clearly has image the ideal generated by the
(q-r) x (q-r) minors of U, which are just the elements of one row of Q(q-r).
Since both domain and image of ~ are rank ! torsion free
modules, ~ gives an isomorphism of (ArM)/t(Ar M) with this ideal. //
The projection of M onto M/tM induces an epimorphism ArM + ~r(M/tM) and hence p(M) = o(M/tM).
If N is another finitely generated R-module of
rank S, then p(M @ N) is the ideal class of (Ar+s(M $ N))/t(Ar+s(M @ N)) = ((ArM)/tOkrM)) ~
((AsN)/t(AsN)) , and so o(M @ N) = p(M).p(N).
Thus p
is a homomorphism from the semigroup of finitely generated R-modules
(with
respect to direct sum) to the semigroup of ideal classes (with respect to product of ideals).
Moreover if F is free p(F) = R, so o(M) depends
only on the stable isomorphism class of M (
p(M ~ F) = p(M)
) and if
P is projective 0(P) is the class of an invertible ideal.
Let M
= HomR(M,R) = HomR(M/tM,R).
torsion free and rank M
*=
rank M = r.
given by~f|A..Afr)(m|A..Amr) if R is a field.
Then M
is finitely generated,
There is a natural map 6:~
= det~fi(mj~,
r(M*) + (~rM)*
which is an isomorphism
The definition of ~ is compatible with localization and
with passage to a quotient with respect to an ideal.
Therefore by
Nakayama's le~m~a ~ is an epimorphism for each prime ideal p of R. P Hence 6 is an epimorphism and therefore 0(M ) = p(M) .
39
The column ideal class of M = coker Q is the row ideal class of coker Qtr, the cokernel of the transpose of Q.
The relationship
between
M and coker Qtr is rather obscure, but there are a pair of exact sequences tr 0 § M * § Rq
(!)
Q
~
Rp § coker Qtr § O
and similarly (2)
0 ~ (coker Qtr)* ~ R p
Q ~ R q ~ M ~ O.
If Q2 is another presentation matrix for M, then it may be related to Q by "Tietze moves" the presentations isomorphic.
[43] and on examining
the effect of such moves on
tr it may be seen that coker Qtr and coker Q2 are stably
If Q is a monomorphism,
then coker Qtr = elM and ~(M) = R.
so that M has a short free resolution, More generally if p.d.M ~ ! then
y(M) is invertible.
Suppose now that M is projective Then the sequence
(2) a b o v e s p l i t s ,
and t h e s e q u e n c e (1) i s s p l i t (since
(1) s p l i t s )
= 0(M) .
modules are reflexive ideals
are reflexive.
f o r any f i n i t e l y of Er(M),
(M =M
and h e n c e c o k e r Q t r = K
exact.
Therefore
Similarly ),
and let K = ker Q = (eoker Qtr)*.
~(M ) = 0(M), s i n c e p r o j e c t i v e
or in particuIar
because invertible ~79]
shows t h a t
g e n e r a t e d module M t h e p r o d u c t 0(M).y(M) i s t h e c l a s s
the ideal generated by .... all the (q-r) x (q-r) minors of Q.
generated module.
true that y(M) = p(M)
If R = Q [ x , y , ~
for every finitely
and M is the ideal
y(M) = p(M) = the class of M, which is not principal, = R.
a l s o pg~er
~(M) = 0(K ) = p(M )
A s i m p l e a r g u m e n t due t o S t e i n i t z
It is not generally
p(M)
is
It would he interesting
(x,y,z) then but M
= R and
to have a simple general characterization
of the column invariant of a module and to find other invariants isomorphism classes of (not necessarily
projective)
modules.
of stable
CHAPTER IV
THE
CROWELL
Let L be a W-component let * be a basepoint homology
for X.
EXACT
SEQUENCE
l-link with exterior X and group G, and The long exact sequence of equivariant
for the covering of the pair (X,*) determined by the maximal
abelian cover p : X' + X gives rise to a 4-term exact sequence of A-modules 0 § HI(X;A) § HI(X,*;A) As Crowell showed that knowledge
§ Ho(*;A) § Ho(X;A) § 0.
of this exact sequence was equivalent
to knowledge of the second commutator quotient study of @ / G " sequence~
could be linearized)
this sequence,
(so that the
we shall call it the Crowell exact
In this chapter we shall establish
Strauss and Traldi relating
~/G"
the elementary
some results of Crowell,
ideals of the members
of
and we shall sketch some results of Massey on the I-adic
completion of the sequence. of Crowell on annihilators
We shall first give new proofs of results and ~ - t o r s i o n
extend to the many component
in knot modules, which
case, and we shall consider when the
A-module G'/G" admits a square presentation matrix. Definition
The Alexander module of L
The k th Alexander polynomial
is the A -module A(L) =HI(X,*;A).
ideal of L is Ek(L) = Ek(A(L))
of L is Ak(L) = Ak(A(L)).
only defined up to multiplication
and the k th Alexander
(The Alexander polynomials
are
by units.)
The right hand map of the Crowell exact sequence may be identified with the augmentation homomorphism hand member is just HI(X' ; ~) transformation;.
considered
By the Hurewicz
s : A + ~ , while the left
as a A-module via the covering
theorem HI(X' ; ~)
is isomorphic
to G'/G"
41
as an abelian group.
There is an alternative description of the
Crowell exact sequence in terms of the link group which we shall give as we shall use it in the next chapter. Let G be a finitely generated group and let eG : ZZ[G] § the augmentation homomorphism of the group ring of G. be the augmentation ideal, generated by { g - l ] g the ideal generated by { w - I ]w in G' } . (g- I) + ~ 2
~=
ker eG
in G } , and let ~ 2 be
The map sending gG' to
gives an isomorphism of G/G' with ~ / ~ 2 .
module of G, A(G) = ~ / ~ 2 ~ '
Let
Zg be
The Alexander
is in a natural way a ZZEG/G' ]-module,
which is finitely generated and hence finitely presentable. conjugation action of G/G' on G'/G",
The
given by gG'.aG" = g ag-l.G '' for
g in G and a in G', makes G'/G" into a 2Z[G/G']-module,
and there is a
natural monomorphism 6 : G'/G" -> A(G) sending aG" to (a- 1) + ~ 2 ~ which has image ~ 2 / ~ 2 ~ .
0
The 4-term exact sequence
~ ~ G'/G" ---+ A(G) ---+ ZE[G/G' ]
(where ~ sends ( g - I ) + ~ 2 ~ sequence for G.
'
to
gG'-I)
~G/G' -+ Zg ---+ 0
is called the Crowell exact
Crowell showed that if f : H § K induces an isomorphism
on abelianization,
then the induced map on short exact sequences of
groups I ---+ H ' / H "
~ H/H"
~
H/H'
---+ I
is an isomorphism if and only if the induced map on the Crowell sequences of 7Z[H/H']-modules
is an isomorphism [37].
Several other
interpretations of A(G) and of the Crowell sequence are given by Crowell
[41], Gamst [53]
and Smythe [175]
.
Thus the Crowell sequence may he written as g 0 ----> G'/G" ----+ A(L) ----+ A
> ~
----+ 0
(I)
42
and is equivalent
to the short exact
sequence
0 ---+ G'/G" ---+ A(L) ----+ I where
I
= ker e = (t I - l , . . . , t
A presentation presentation
matrix
for A(L)
0
,~
(I)'
- l) is the augmentation
= A(G(L)) may be obtained
for G(L) via the free differential
shall be contemt with the information
calculus
obtainable
ideal of A .
from a [43]
, but we
through homological
algebra.
Rank, Projective
Dimension
Since X is a compact bounded it is homotopy 1 0-cell,
equivalent
n + ! l-cells
3-manifold
to a finite
and n 2-cells,
of X' is chain homotopy
equivalent
and ~ - torsion
with Euler characteristic
2-dimensional
so the equivariant
(X',p-l(*)) ...
Since Hq(X,*;A)
0
Definition
is then chain homotopy
> 0 ---+ D 2 ---+ D I = Hq(X;A)
> H2(X;A)
The Alexander
D, with
The relative
equivalent
complex of
to the complex
~> 0 .
for all q ~ 2, there
~
chain complex
to a finite free complex
D o = A, D I = A n+! , D 2 = A n and Dq = 0 for q > 2. the pair
cell complex with
is an exact
sequence
An ---+ d An+l ---+ A(L) ---+ 0 .
nullity
(2)
of L, ~(L), i8 the rank of A(L)
as a
A-module. It is i m e d i a t e complex
obtained
(X',p-l(*)) so A(L)
~) ~
that ~(L) = min{k]Ek(L)
by tensoring
over A with ~ ~HI(X,*
;~)
the cellular
# 0}
chain complex
is just the cellular = ~.
and is
Therefore
~I.
of the pair
chain complex
~(E
The
(L)) = ~
of (X,*),
and so
0,
43
~(L) ~ ~.
The Crowell sequence implies that rank G'/G" = ~(L) -i and
tG'/G" = tA(L), while the exact sequence (2) implies that H2(X;A) is torsion free of rank ~ ( L ) - I.
Lemma I (Cochran [ 29]) Proof
If ~(L) = 2, then H2(X;A) ~- A.
Let u and v belong to H2(X) ~ A n .
there are ~ and ~ in A such that ~u = By. no common factor.
Then since rank Hi(X;A) = l, We may assume that ~ and B have
Since fi is factorial v = ~w for some w in A n , which
must actually be in H2(X;A) by the exactness of is torsion free.
(2) and the fact that A n+l
Therefore every 2-generator submodule of the finitely
generated rank 1 A-module H2(X;A) is cyclic.
The lemma follows easily. //
Cochran's result extended to embeddings of arbitrary finite graphs and was published in [ 3 0 ] .
In general H2(X;A) is free if and only if
the projective dimension of A(L) is at most 2.
For if p.d. A(L) ~ 2 ,
Schanuel's lemma applied to the exact sequence (2) implies that H2(X;A) is projective, and Suslin has shown that every projective A-module is free [185].
The argument in the other direction is obvious.
For a
boundary link a Mayer-Vietoris argument shows that H2(X;A) is free of rank ~ - I, but the 3-component homology boundary link of Figure V.! has A(L) ~ A S (~ (A3/(t I - 2 , S not free.
t2 + !
t3 - I ) )
and so for this link H2(X;A) is
If L is unsplittable X is an Eilenberg-MacLane space K(G,I)
for the group G (by the Sphere Theorem) and then Hq(G'; ~) = Hq(X;A). Thus in particular the commutator subgroup of a classical knot group has trivial integral homology in degree greater than I. The Crowell exact sequence for the free group F(~) is 0 ----+F(B)'/F(~)" ---+ A ~ ---+ A ---+ ~
---+ 0 .
The right hand terms constitute a partial resolution for the augmentation
44
module
~.
We may obtain a complete
equivariant
homology
of (SI) ~.
The latter
of ]R~ , considered
the lifts to ~
vertices
all have integral
cellular
are the Euclidean coordinates.
Since
covering
space
with
(~) q
of side ! and whose
]R ~ is contractible~
sequence
(q ).
with Cq free of rank
we may obtain
copies
from the
structure
q-cubes
0 --+ C~ ---+ C~_! ----+ ... ---+ C I ---+ C O
explicit,
for ~
as the universal
space has a natural
q-cells;
there is an exact
A -resolution
Alternatively,
the complex
of the corresponding All the differentials
~ 7z ---+ 0
to make
(3)
the maps more
(C,) as the tensor product
complex
over 7z of
t-! for S I : 0 --+ A I ---+ A I --+ 0.
of the complex
(ZZ ( ~ A C,) are 0, so
(V) TorA(zg, ZZ) eP2Z
= 97.
sequences
= Hq((SI)~; ZZ) .
(These
I, and of Poincard
presentation
Lemma 2
(i)
for F(V)
with
(v) 2
generators
form
and
= u-l
< p.d. G'/g"
= ~-2
o._r
p.d. A(L)
= p.d. G'/G"
>. p - I .
from the exact sequence
assertion
of e (-) = ExtA(-,A ) applied
(I)',
the Crowell has a
- 2}
p.d. A(L)
The second
of the spectral
F(V)'/F(~)"
o__r
lemma.
and
and the contractability
< p.d. A(L)
follows
= 0 if q # V
( v ) relations 3 "
p.d. H2(X;A ) = max{0,p.d.A(L)
The first assertion
Schanuel's
eqTz
(3) together with
imply that the A-module
(ii) Either p.d. G'/G"
sequence
duality
Note also that the resolution
sequence
Proof
Moreover
are also iu~nediate consequences
of Chapter
of IRV). exact
= ZZ q
follows
from the long exact
to the Crowell
and the fact that e q-I I = e q ~
(2) and
exact
for q > 0. //
sequence
in the
45
An immediate
consequence
of this lemma is that if AI(L ) # 0
then G'/G" has a square presentation matrix if and only if ~ ~ 3.
For
if a(L) = ] then p.d. A(L) ~ 1 and G'/G '~ is a torsion module of projective dimension ~ - 2
unless p.d. A(L) ~ p - I .
Since projective
free, a torsion module has a short projective
resolution
A-modules
are
if and only if
it has a square presentation matrix. If ~ known.
= I, then ~(L) must be 1 also and the result is well
If
~
= 2, the module G'/G" has a square presentation matrix
even if ~(L) = 2. modules
This was proven by Bailey who characterized
arising from 2-component
links as the A2-modules
square presentation matrix of a particular form [ 7 ] [34 ] and Chapter VII).
such
admitting a
(See also Cooper
It may also he seen as follows.
Let
Z ~ D 1 = A n+l be the submodule of l-cycles in the cellular chain complex of X.
lq~en there are exact sequences 0 ---+ H2(X;A) ---+ A n---+ Z ---+ G'/G" ---+ 0
and
0
By Sehanuel's
> Z ~
A n+l ---+ A ---+ ~
lemma Z is projective,
hence free, and clearly rank Z = n.
Hence G'/G" has a square presentation matrix. p.d. G'/G" ~ 2 since H2(X;A) ~(L) = 2) by Cochran's
Theorem 3 nullity a.
---+ 0.
Note also that
is either 0 (if a(L) = I) or free (if
lemma.
Let G be the group of a ~-component
link L with Alexander
Then
(i)
if ~ = ], El(L) is principal,
(ii)
if E _I(G'/G")
while if p ~ I, El(L) = (AI(L))).I;
an_d E (L) are both principal,
(G' = G") then ~ ~ 2;
or if G' is perfect
46
(iii) if e = ], then G'/G" has no nontrivial pseudozero submodule and Ann(G'/G") =(AI(L)/A2(L)); (iv)
for each k ~ I, ek(tG'/G") ~ = (A~+k_] (L)/Ae+k(e)) 9
Proof
(i)
We may clearly assume that El(L) ~ 0, so that A(L) has rank I
and A ( L ) / t A ( L ) ~
I.
By Theorem 111.12 the isomorphism class of this
ideal is just the Steinitz-Fox-Smythe row invariant of A(L).
From the
exact sequence (2) we see that the column invariant is the class of the principal ideals, and Steinitz showed that the product of the row and column invariants was the class of the first nonzero elementary ideal [179].
The assertion follows readily. (ii) If E _I(G'/G") and E (L) are both principal, then the Crowell
exact sequence gives rise to a projective resolution of ~
of length 2
0 ----+ (G'/G")/(tG'/G") ---+ A(L)/(tG'/G") ---+ A ---+ ~ Hence ~ = p.d. ~
~ 2o
Similarly, if G'/G" = 0 then ~ = !
~ 0. so
p.d. A(L) ~ I and A(L) = I so p.d. ~ $ 2. (iii) If ~ = | then p.d. A(L) $ I and G'/G" = tA(L), so the assertions follow from Theorems III.4 and III.10 and the Remark after Theorem III.10. (iv)
This is a consequence of the Corollary to Theorem III.4.//
Part (i) of this Theorem was first proven by Torres who used properties of the Wirtinger presentation
[189].
If the commutator sub-
group of a 2-component link is perfect, then AI(L) = I, so the linking number is •
by the second Torres condition.
the linking number is for G/G 3 of Chapter I.
•
(See Chapter Vll.
That
also follows from the Milnor presentation See also Chen [27]).
In the knot theoretic
47
case (~= ]) the results of part (iii) were first obtained by Crowell [39 ] (Note that a Al-mOdule is pseudozero if and only if it is finite).
In
[40 ] he showed that AI(L) annihilates G'/G" (under an unnecessary further hypothesis).
The knot 946 [ 157 ;page 399]
G'/G" = (Al/(t- 2)) (~) (Al/(2t-I)) (3,t + I) is not principal.
has
and so ~2(G'/G") = ( t - 2 , 2 t - I )
=
The argument of our next theorem is related
to that of Crowell in [39 ]
Theorem 4
Let M be a finitely senerated A-module of rank r such that
Er(M) is principal and suppose that C(Ar(M)) = •
Then M is torsion
free as an abelian group. Proof
Let p be an integral prime and suppose m is an element of M such
that p.m = 0.
Then Ann(fi.m) contains p and Ann(tM), and hence Eo(tM)
by Theorem Ill.l, so if Ann(A.m) ~ = (6), ~ divides p and Ao(tM) = Ar(M). Since e(Ar(M)) = •
~ must be
•
and so A.m is pseudozero.
It now
follows from Theorems ]]1.9 and III.I0 that m = 0 . / / The condition e(Eo(M)) = ~ the case B = I, r = 0 and ~
is equivalent to ~
~A
M = O.
For
(~A M = 0, Crowell proved this Theorem
under the additional assumption that M has a square presentation matrix [ 3 9 ] , Levine proved that this additional assumption is equivalent to such an M being torsion free as an abelian group [ 119], and Weber has shown that for such an M these conditions are also equivalent to Eo(M) being principal
[202].
When L is a 2-component link a little more can be said about ~ - t o r s i o n in G'/G".
If At(L) # 0 then G'/G" has nontrivial p-torsion
for p a prime integer if and only if p divides AI(L) , in which case p must divide the linking number AI(L )(1,1) (by the second Torres condition).
48
For Ann(G'/G")
is generated by AI(L)/A2(L) , which is divisible by each of
the prime factors of AI(L).
Levine has shown that given any X in A 2
such that X = ~ there is a 2-component
link L with linking number 0 such
that AI(L) = X(t I - l)(t 2 - I) Ill7 ].
Hence on taking X to be a prime
integer we see that G'/G" need not be torsion free as an abelian group. If AI(L) = 0 and G'/G" has ~ - t o r s i o n Theorem 3.
then E2(L) cannot be principal,
However since p.d. G'/G" $ 2, as remarked
contains no nontrivial
finite A-submodule,
by
above, G'/G"
by Remark 2 after Theorem III.ll.
Link module sequences
his work on the A-modules A(L) and G'/G" Crowell defined a link
In module
sequence as an exact sequence 0 ---+ B ---+ A ---+ I ---+ 0
of A-modules
(4)
such that A has a presentation with n + I generators
n relations
(for some n) and where I is the augmentation
ideal of A.
arguments of Theorem 3 apply in this slightly more general showing for instance (i)
that El(A) = (AI(A))I.
The first elementary
In [40]
and
setting,
Crowell showed that
ideal E l of A annihilates
B.
If E l #0,
then B is the torsion submodule of A; (ii)
If the product
Alexander polynomial
I) does not divide the by AI;
(as is the case with the Crowell exact
(I)' of a link) (iii) The sequence (iv)
e(AI(A))
... ( t -
A 1 of A, then B is annihilated
while if also ZZ O A A = ZZ~ sequence
(t I -I)
=
(4) never splits if ~ >, 3;
If ~ = 2 the sequence
+ I.
The
(4) splits if and only if
49
The results (i) and (ii) are contained in part (iii) of Theorem 3, while the other results follow on tensoring the sequence (4) over A with ~. For the exact sequence can only split if ~ O
I.
A = ~
=
Therefore we may assume AI(A) # 0, since otherwise E o ( ~ Q
B)
would be 0.
~) B = 0, as ~
~
From this and the other assumptions on A it follows that A The long exact sequence of T o r , ( ~ , - )
Tor~'(Z~, ^ A) = ZZ~-I .
applied to
(4) then shows that rank Zg @
B
~
rank TorlA(Zz) I) - rank TorlA(Tz,A)
which is greater than 0 if ~ >~ 3.
=
(~) 2
If B = 2 the module ZZ O
- ~ + I B has a
square presentation matrix with determinant generating E(Eo(B)) and so divisible by C(Ao(B)) = e(Al(A)) , and thus 2Z O
B = 0 only if
e(Al(A)) = _+ I. Crowell showed also that (if ~ = 2 and EXtA(B,I) ~
7Z/e(AI(A))
and asked whether the class of the extension
could be used to distinguish That Z g ~
A 1 (A) # 0)
between two links.
B # 0 whenever D >~ 3 follows also from a result of
Crowell and Strauss who showed that for any link module sequence (~22) Eo(B) = (AI(A)).I
[44].
This was rediscovered by Bailey [ 7 ] and
extended by Traldi [190] who showed that (a)
Ek(A) _~ Ek_ !(B).I ~-I
(b)
Ek_ l(B) ~ Ek(A).I
for all k;
Q21 ) for all k;
(lJ21) +k-~ (b')
Ek_l(B) ~ Ek(A).I
(Here if p < 0, Ek(A).IP _C Ek_|(B)
for ! .< k .< ~. means Ek(A) C Ek_I(B).I-P) .
Note
50
(~2) that (b') implies Eo(B) ~ (AI(A)).I
, since El(A) = (&I(A)).I, thus
proving part of the Crowell-Strauss result.
We shall sketch a proof
of (a) and of part of (b). On applying lemma 111.6 to the link module sequence (4) we see that Ek(A ) ~_ Ek_I(B).EI(1)
for any k, and it is not hard to show by
induction on ~ that El(l) = I ~-1, using the presentation C 2 § C I + I § 0 derived from the A-resolution for ~ Jacobian presentation
given above.
(This is also the
for I = A ( ~ ~) obtained via the free differential
calculus from an obvious presentation for the group ~ Lemma 5.2 of [44 ].)
.
See also
This proves (a).
The link module sequence (4) together with the Crowell sequence for F(B) gives rise to another short exact sequence 0
~ F(~)'/F(~)" ---+ B (~) A ~ ---+ A
~ 0.
Hence Ek_I(B) = Ek+~_I(B~)A~) ~ Ek(A).E _I(F(~)'/F(~)") for any k ~ I. Thus (b) is true in general if and only if it is true for the Crowell exact sequence
for F(~) when k = ~.
(This special case is established
in Le~na 5.6 of [44].)
Completion of link module sequences
Stallings' theorem implies that the nilpotent quotients G/G"G n = (G/Gn)/(G/Gn)" of a link group are invariant under (possibly wild) 1-equivalence of the link.
These quotients have been called the
Chen groups of the link by Murasugi, who used free differential calculus to show that, if ~ = 2, these groups are "free" if and only if E~_I (L) = 0 if and only if the longitudes of L are in G(=) = n~>,2(Gn G'') [140].
In the next chapter we shall give a new proof of this result,
51
applicable to links with any number of components, and we observe that the Alexander nullity of a link is invariant under 1-equivalence. Massey has also extended the first equivalence of Murasugi's theorem, using conmautative algebra in a similar but more whole-hearted way than we do E125].
Although he obtains other interesting results, our
mixture of commutative algebra and group theory seems necessary to derive the condition on the longitudes, and so we shall only sketch proofs of some of his results here. He observed that the nilpotent completion of G/G" corresponds to the l-adic completion of B = G'/G", since B/InB = G'/G"G +~, and so n g
considered the l-adic completion of a llnk module sequence (4) such that 0
A = ~
.
(Since completion is an exact functor, the completed
sequence is also exact.)
Let M denote the I-adic completion of a
A-module M, so M is a A-module, via t i >-+ I + X i.
and A embeds in fi ~ E [ X I
Then he proved the following theorems.
..... X ]] (We have
changed the notation and abbreviated his enunciation slightly.) ^
I
The A-module A has a presentation with u generators and s < relations.
II
The A-module B has a presentation with ( ~2)generators and (3~) + s relations.
Moreover (3 ) of these relations are the same for all
p-component links. III If ~ = 2 the associated graded module G(B) = G(B) is a cyclic module over G(A) = G(fi) = ~ [ X 1 ..... Xn] , and Ann G(B) is generated by the "initial form" of the image of the Alexander polynomial AI(A) in ft. Thus the Chen groups of a 2-component link are effectively determined by its Alexander polynomial. (Here the initial form of a power series of fi is the homogeneous
52
polynomial in X. consisting of the nonzero terms of lowest degree.) i IV
The completed Crowell exact sequence 0 ---+ G'/G" ---+ A(L) ---+ I ---+ 0 of a link is invariant under 1-equivalence (hence under isotopy and concordance).
Corollary
The principal ideal in the power series ring ~ generated by the
image of the Alexander polynomial is an invariant of the link under I-equivalence. The first two theorems follow from the link module sequence and the standard presentation of I, on using Nakayama's lemma, and the third is a consequence of the second, together with a little group theory. Massey observes that if ~
= AI(L)(I,I) # 0 then the initial form of
AI(L) is the constant ~, while if % = 0 the Torres conditions (see Chapter VII) imply (a)
the initial form of AI(L) is an homogeneous polynomial of even degree in X I and X2;
(b)
n n if the initial form has degree n, the coefficients of X 1 and X 2
are
both 0. He asks whether these characterize such initial forms, and verifies that they do for n = 2, and that any even degree can occur.
The fourth theorem
is a consequence of Stallings' theorem, while the corollary follows from the fact that the principal ideal generated by the image of AI(L) in A is the ideal (~I(~(L))).
This corollary may be restated in the following
form, derived earlier by Kawauchi
[89 ].
$3
Corollary
Let AI(L) = 61.u I
augmenting to
• I.
where c(u I) =
• I and ~I has no factor
Then ~I is invariant under I-equivalence.
For an element of A becomes a unit in A if and only if it augments to (Kawauchi's argument applies only to PL I-equivalences,
• I.
as it assumes
that the equivariant chain complexes of the maximal abel•
cover of the
complement of the I-equivalence are finitely generated.)
We may now give a simple proof of the following theorem.
Theorem 5. then
Let
G = i, ~
Proof.
If
or
L
assume that
L: ~S 1 ~ S 3
= 0. and
InB = 0
If
G
is nilpotent,
has nilpotent group then so do all its sublinks. p ~ 3.
It will suffice to prove that
G' = G 2 ~ G 3 ~ G", As
G.
~2.
Since these are true if Since
be a link with group
G
~ ~ I,
n
we may assume that
large,
B so
and that
~ ~ 2.
it will suffice to prove that
is nilpotent, for
G2 = G 3
So we may
Let
B=]]3,
B = G'/G". i.e. that
is finitely generated as an abel• B = B.
~ ~ 2.
group,
Massey's Theorem II implies that
^
has deficiency an abel• ~
9
Since
> 0
as a A-module,
group if it is H2(G;~) = ~ (2) ~
0.
and so can only be finitely generated as
Therefore
G
is a quotient of
is abel•
and so isomorphic to
H2(X;~) = ~ - I
[83],
~ ~ 2.
(Using the Sphere Theorem and the Loop Theorem one can in fact show that if a link has solvable group then the link is empty,
O0
.)
the unknot or
//
CHAPTER V
THE VANISHING OF ALEXANDER IDEALS
At the 1961 Georgia conference on Topology of 3-Manifolds, Fox raised the question of the geometric significance of the identical vanishing of the first Alexander polynomial of a 2-component link [50; Problem 16~. Boundary links clearly satisfy this condition, but in 1965 Smythe introduced the concept of "homology boundary llnk" to show that such a link need not be boundary [17~.
He conjectured in turn that "AI(L) = 0"
should imply that L be an homology boundary link.
In 1970 Murasugi proved
that this condition is equivalent to "each of the subquotients G"Gn/G"Gn+ 1 is isomorphic to the corresponding subquotient of F(2) " and to "the longitudes of L are in G(~) =
N (G"G n) "[140]. n~2
Cochran showed in his
1970 Dartmouth thesis that "AI(L) = 0" implied that H2(X;A ) =
A, and
constructed a family of unsplittable 2-component links with first Alexander polynomial 0 [29, 30].
In this chapter is given a counter-example to the conjecture of Smythe, as an illustration of a new criterion for a ribbon link to be an homology
boundary llnk.
The more general situation of the vanishing
of certain of the Alexander ideals of a finitely generated group with abelianization ~
is considered, and it is shown that the rank of the
Alexander module A(G) depends only on the nilpotent quotients of G/G". As a consequence the Alexander nullity of a link is an invarlant of arbitrary 1-equivalence.
Conversely if ~(G) = ~ then any map of F(~)
to G inducing an isomorphism on abelianization induces isomorphisms on all such nilpotent quotients.
Furthermore if G is the group of a
~-component link L, then ~(L) = ~ if and only if the longitudes of L
55
lie in G(~).
We shall also answer a question raised by Cochran (for
2-component links), by showing that H2(X;A ) = H2(X';~ ) projects onto H~X;~)
if and only if E _I(L) = O.
The Counter-example
An epimorphism of groups f:G + H induces an epimorphism A(f):A(G) § A(H). Therefore if G is finitely generated, so the elementary ideals E,(G) = E,(A(G)) C ~ [G/G'] are defined, the image of El(G) in ~[H/H'] is contained in El(H).
Now let H be a group with a presentation of
deficiency ~ and with abelianization H/H' = ~ . i < ~ and E (H) ~ (1) modulo I.
Then E.(H) = 0 for i
Hence if L is an homology boundary
link, so G(L) maps onto H = F(~), then E _I(L) = O.
By Theorem 11.6
this is also true of ribbon links (taking H = H(R)) and so they provide examples on which to test Smythe's conjecture.
Since any epimorphism
G(L) + F(~) induces an isomorphism G / G ~ F(~), it must s H(R) if L bounds a ribbon R:~D 2 + S 3.
through
Thus the criterion of the next
theorem may suffice to show that a ribbon link is not an homology boundary link.
Theorem ! H/H' = ~ ,
Proof
If H is a group with a presentation of deficiency ~ and with which maps onto F(~)/F(u)", then E (H) is principal.
The assumptions
Therefore A ( H ) ~
imply that A(H) has rank B and maps onto A .
A ~ (~) tA(H).
Since A(H) has a presentation of deficiency
U, adding ~ relations to kill a basis for the free s,mmand gives a square presentation matrix for tA(H). is principal.//
Therefore E (H) = E (A(H)) = Eo(tA(H))
56
In the next chapter we shall give several partial converses of this theorem.
We shall now present our first counter-example
to Smythe's
conjecture.
1 Figure 1
The solid link in Figure 1 (which has unknotted components) to a ribbon map R with 4 singularities.
extends
The ribbon group H(R) for this
ribbon has a presentation {Xl,X2,XS,Yl,Y2~Y3lyTlxlYl
-1
= x2,ysx2y 3
which is Tietze-equivalent
-1
-1
= Xs,X ~ ylxl = y2,x3 y2x3 = y3}
E43; page 4 4
to
{ X l , X S , Y l [ X l y T l x l Y l X l-1 = ( x 3 x T l ) - l x T l y T l x l ( x 3 x T l ) y l x l ( x 3 x T l ) } so H(R) has a preabelian presentation {x, y, a I xY -I xY x-I = a-I x-I Y-I x a y x a } The Jacobian matrix of this presentation
is
M = [I (Y-I)(x-ly -I - xy-l),(l-x)( x-I Y
-I - xy -I
-I
57
ans so
E2(H(R)) = ((y-l)(x2-1), = (x+1, y-l-xy) = (x+l,
since x-I = y
-I
(l-(y-l-xy))
2y-l)
which is clearly not principal. afortiori,
(l-x)(x2-1), y - ] - xy)
Thus H(R) cannot map onto F(2)/F(2)";
the group of the link cannot map onto F(2).
We shall give several other proofs that this link is not an homology boundary link in Chapters VI and VIII.
(Note however that on
removing the half twist from the lower ribbon we obtain Milnor's boundary link ~30; page 305]).
In Chapter VI we shall also give an example of a
ribbon link which is not an homology boundary link although G/G" ~ F(2yF(2)", so that this cannot be proven using Alexander ideals.
The link of Figure II.I is a ribbon homology boundary link for which E2(L ) = (x-l, y2-y+l), so E (H(R)) principal need not imply E (L) principal.
This is the simplest nonsplittable link with Alexander
polynomial O, as may be seen from the tables in ~57].
(Smythe's original
homology boundary link may be obtained by giving the knotted ribbon of this link 3 half twists).
It is not hard to verify that any ribbon counter example
to Smythe's conjecture must have at least 4 ribbon singularities.
The
examples constructed by means of Baumslag's parafree group in Chapter II show that the higher dimensional analogue of Smythe's conjecture is false. It is clear that for these groups El(G) = O, but more generally E _](G) = 0 for G the group of any
b-component n-link
(for n ~ 2) as follows from
Stallings' theorem and the theorem of the next section.
$8
Alexander Ideals and Chen Groups
In [|40] Murasugi proved, inter alia, that for G the group of a 2-component link the following conditions are equivalent (I)
El(G) = O;
(2)
the Chen group
Q(G;q) = G G"/G G" is isomorphic to Q(F(2);q) q q+l
for all q ~ l; (3)
N (GqG") " the longitudes of the link are in G(~) = q~1
In the course of his proof, which involved delicate computations in the free differential calculus, he found presentations for the (finitely generated abelian) groups Q(H;q) for H free of finite rank and for H the group of a 2-component link.
In this section we shall give a proof of the following generalization (which was published in ~2] ).
Theorem 2
If G is a finitely generated group with GiG' ~ ~ t h e n
the
following two conditions are equivalent:
(I)
E _l(G) = 0
(2)
Q(G;q) ~ Q(F(~);q) for all integers q ~ I.
;
Furthermore, if G is the group of a ~-component link L then (I) and (2) are equivalent to (3)
the longitudes of L are in G(~).
Instead of calculating free derivatives, we shall use Nakayama's lemma and Krull's theorem, applied to the Crowell exact sequences for the groups G/G .
(As indicated in Chapter IV, Massey has also used
c o m u t a t i v e algebra to extend Murasugi's equivalence (1) ~=#(2) to arbitrary link groups, and he has shown that the Chen groups of a
59
2-component link may be effectively computed from the Alexander polynomial [12~).
For any group H, let H = H/H". naturally isomorphic.
Then H/H H", H/(H)q and ~ q
q
are
A mapf:H-~K induces maps f : H - ~ K and f :H/H -4 K/K , q q q
and (~)q is naturally equivalent to (fq).
We shall identify these naturally
isomorphic quotients of H and naturally equivalent maps.
There are short
exact sequences I + Q(H;q) + H/Hq+IH" § H/HqH" + I by definition of the Chen groups, and so by the five lemma and induction a map f:H § K induces isomorphisms on all Chen groups if and only if all the maps fq are isomorphisms.
The arguments below will be in terms of the groups H/HqH"
excepting for one appeal to the computation of the Chen groups of a free group by Chen and Murasugi.
The qth truncated Alexander module of H is the ~[H/H'~-module A (H) = ~ / ~ 2 ~ + ~ q ; q
in ~articular A2(H) = ~/~2 is isomorghic to H/H'
(see Chapter IV).
Given a finite presentation for H and an isomorphism
H/H' = ~ U , the Jacobian matrix of this presentation at this map is a presentation matrix for A(H) over A
and this matrix reduced modulo
Iq-! is a presentation matrix for A (H) over A/I q-l [175]. Hence q A q (F(~)) = (A/lq-l) ~. each q ~ I.
Let ~ be the ideal generated by {w-llw 6 H q }, for ~q
Then since
Ex,h~-I
follows by induction that ~q ~ q and Aq(H/Hn) = ~ / ~ 2 ~ + ~ n
+~q
= ((x-l)(h-l)
-
(h-l)(x-l)
for each q ~ ]. [175].
x -I
h -l
it
Then A(H/H n) = r
If f:g § K, let A(f) and Aq(f) be
the maps induced on A(H) and A (H) respectively. q
In particular, the
quotient map p:H § H induces isomorphisms A(p) and Aq(p).
60
Proof of the Theorem 82: ~
Choose a map 8:F = F(U) + G inducing an isomorphism
+ G/G', and hence an identification of A = A
with ~[G/G'].
a map shall be referred to below as a "meridian" map).
(Such
Let ~ : ~[F~ § =[G]
be the induced map of group rings. (I) implies (2). R/I = ~,
Suppose E _I(G) C Iq-1.
Let R = A/I q-l.
since the image of I in R is nilpotent and ~
Since Aq(8) ~ R ~
Then R/rad R =
is a domain.
= 82 is an isomorphism, the map Aq(8):R ~ = Aq(F) § Aq(G)
is onto by Nakayama's lemma E4; page 213 . The kernel of this map is finitely generated, since R is noetherian, and so A (G) has a presentation q Ra M R ~ A (G) § O. q Since E
!(Aq(G)) = E _](G) reduced modulo Iq-] = O, the matrix M must be
null, and so Aq(8) is an isomorph{sm.
Thus if E _](G) = O, the maps
A (8) are isomorphisms for every q. q
By the Crowell equivalence of Chapter IV, to show that the maps q are isomorphisms (and hence that the Chen groups are isomorphic), it will suffice to show that each A(eq):A(F/Fq) + A(G/Gq) is an isomorphism. Since 82 is onto, 8q:F/Fq § G/Gq is onto ~23; page 350~ and so A(eq) is onto.
On considering (for each r) the commutative diagram
A(F/Fq)
A (F/F) r q
A(G/Gq)
~ Ar (G/Gq)
in which each map is onto, it will suffice to show that the map Ar(F/F ) -> Ar(G/G q) q is a monomorphism for each r, and that
r>~!Oker(:A(F/Fq) § Ar(F/Fq)) = O. The map
61
At(0): / / / 2 ~
+ /r
ker ~ C ~ - I ( ~ 2 ~ ~2~
, 2/~2 ~
+~r)=
+ 9 r + ~'
/2/
+ ~r +/r.
is an isomorphism, so Clearly [ ( ~ /
+ /r
+ ~q)C
and since 8:F § G is onto modulo G r, it follows that
~s C_.~ ( ~ s ) + ~ r
for each s.
(For if gs is in Gs, then gs = e(fs)gr
for some fs in Fs and gr in G r by induction on s, and hence gs -
O(fs)g r
- | =
O(f s)
- | +
+ ~r)
ArCF/F q) = ~ / t 2 #
- I) is in ~ ( / S ) + > r C
=
~'(~S)
+
+ ~ q C ~2 V + ~ r + ~(~q) and so ~-I 8 (~2~
Therefore V 2 ~ + ~ r
~-l(~2~
8(fs)(f r
I
+ ~
= f2/
+ ~r
+
~r). ~r +
~q) =
+ ~ r + ~q, that is, the map from
+ /q
to Ar(G/G q) : 7 1 7 2 7
+ 7 r + ~
is a
monomorphism (and hence an isomorphism). =
Now A(F) = A ~ and A (F) = (A/It-l) ~ r A(F)/Ir-IA(F), so on considering the commutative diagram A(F)
~ A(F/Fq)
A r(F)
> A r(F/Fq)
it follows that ker(: A(F/Fq) § Ar(F/Fq) ) = Ir-IA(F/Fq).
Since A(F/Fq) is
finitely generated over the noetherian ring A, r>~IN(Ir-IA(F/Fq)) = {= in A(F/Fq) I (1+j)~ = 0 for some j in I}
by Krull's theorem E4; pagelIO~.
Now A(F/Fq) sits in the Crowell sequence 0 + F'/FqF" + A(F/Fq) + A § ~ + 0 so if ~ in A(F/Fq) is such that (l+j)~ = 0 for some j in I, then ~ is in F'/F qF".
But F'/F qF" = (F'/F"/Iq-I(F'/F '') is a module over A/I q-1 and
]+j is invertible in A/I q-l, for any j in I. completes the argument for (I) ~
Therefore e = O.
This
(2).
(2) implies (I). Suppose that there is an homomorphism $:G § H which
62
induces an isomorphism
~q: . G/G
q
. . + H/H . q
Ar(H/H q) are isomorphisms for all r. and ~ q
. . . Then the induced maps Ar(G/G q) +
Since Ar(K/K q) = ~ / ~ 2 ~
~ q C ~ r for r 6 q, Ar(K/K q) = Ar(K) if r ~ q.
+ ~r
+ ~q
It follows that
the induced map Aq(G) + Aq(H) is an isomorphism and so E,(G) ~ E,(H)
modulo
Iq-I . Hence if ~ induces isomorphisms on all Chen groups, so all the maps ~q are isomorphisms (as above) the E.(G) = O if and only if Ej(H) = O J (since N I q-I = 0). In particular, if the Chen groups of G are "free", q~l then the maps 0
q
induced by a "meridian" map 0, which are always epimorphisms,
are isomorphisms by induction, the five lemma and the hopficity [123; page 296] of finitely generated abelian groups Q(F;q) (applied to the conanutative diagrams
I
~
1
q(F;q)
+ F/Fq+I F ' '
~ F/FqF"
, 1
G/Gq+IG"
~ G/GqG"
~ I )
~ Q(G;q)
and so E _I(G) = 0 (3) implies (2).
Suppose now that G is the group of a link L.
Milnor's theorem G/G
q
Then by
has a presentation {x., I ~ i ~ ~ I ~i' e ~ i
| ~ i Z ~,
coDanutators of weight q+|} where e. is a word representing the image of the 3 .th j longitude in G/Gq, so if the longitudes are all in G(~), G/GqG" is "free", and a "meridian" map induces isomorphisms of Q(F;q) with Q(G:q) (2) implies (3) Lenmaa Let fl,...,f generate F/F F". q is generated by f Proof
Then the centraliser of f
(c) = O.
in F/F F" q
and F q-~.F"/F q F".
Suppose f c = c f . Let ~.:F/F F" + ~ ~ l q
for ! ~ i, j ~ ~.
U
be defined by ~i(fj) = ~ij
Without loss of generality, it may be assumed that
To show that l.(c) = 0 for all j it suffices to pass to the
63
quotient group obtained by killing f. for all i ~ j,~ and then to F(2)/F(2) 3 1
where it is clear.
So it may be assumed that c is in (F/FqF")'.
[-,f ] induces a I-I map Q(F;r) + Q(F;r+l)
Now
for r ~ 2 (it maps distinct
standard elements of length r to distinct standard elements of length r + l ~25,140~) and so by induction c is in Fq_IF"/FqF" and the lemma is proven. Consequently if the Chen groups of G are "free", then the quotients G/G G" are "free", and since they are generated by the meridians, and q .th since the jth longitude commutes with the j meridian, it follows by induction that all the longitudes are in G(~).
//
The image of the Chen kernel G(~) in G'/G" always lies in tG'/G", since G(~)/G"
=
n^ ( G G l, /G ,, ) = N-In-2(G'/G '') = {g in G'/G" n~Z n n~2
for some j in I}.
If E _I(L) = O, G(~) maps onto tG'/G".
contains some ~ such that e(~) = I, since e(E (L)) = ~.
I (|+J)g = 0 For E (L)
Let S = {~n in ~ O}.
Then A(L) S is a projective As-module by the Corollary to Theorem 111.9, so some power ~N of ~ annihilates
tA(L).
1 + j, where j = ~N _ 1 is in I.
Hence tG'/G" is annihilated by
Conversely if G(~)/G" = tG'/G" and if the
linking numbers of L are all O, so that the longitudes are in G' and hence have image in tG'/G", then they are in G(~) and E l ( L )
= O.
If G has a presentation of deficiency ~ and G/G' = ~ so the Chen groups are "free". then also "free"
~23;
then E l ( G )
In fact the nilpotent quotients G/Gq are
page 353~.
For link groups the nilpotent quotients
G/Gq are all "free" if and only if the longitudes are in G , as may be seen by arguments similar to those for (2) 4=> (3). is the case if the link is 1-equivalent the nilpotent quotients G/G Stallings'
theorem.
q
In particular,
this
to an homology boundary link, for
are invariant under 1-equivalence,
by
If G is a link group with all Chen groups "free", are
all the nilpotent quotients G/G
q
"free"?
This is certainly false for
other groups, for instance the group presented by {x,y I [[x,Y~,[x,Y-l]] }-
= 0
(Is this a link group?)
More geometrically, does the vanishing of
E _I(G) for G the group of a U-component link L imply that L is concordant to a boundary link, or at least 1-equivalent to an homology boundary link?
The argument used to show that (2) implies (I), together with the invariance of the nilpotent quotients under 1-equivalence, actually gives the stronger
Corollary
The Alexander nullity of a link is invariant under 1-equivalence.
In particular if L is a U-component slice link then =(L) = ~.
This corollary is also a consequence of Massey's Theorem IV (see Chapter IV) and has also been found independently by Kawauchi and Sato ~ 6 ~ .
~0 3
(However their proofs apply only for PL 1-equivalences).
A Question of Cochran
A 2-component boundary link L may also be characterized as one for which there is a connected closed surface C in S 3 which separates the components of L and such that each component is nullhomologous in the complement of C.
Such a surface represents a generator of H2(X;~)
lifts to a generator of H2(X';~)
= H2iX;A ) .
and
In attempting to decide
whether 2-component links with first Alexander polynomial O were homology boundary links, Cochran showed that for such links H2(X;A) is a free module of rank I, and asked whether the map to H 2 [ X ; ~ ) induced by the projection p was onto E 2 ~ if El(L) = 0.
(See Chapter IV).
It is clear that this is only possible
In this short section we shall show that this condition is
also sufficient, thereby answering Cochran's question affirmatively, and our argument shall resolve the corresponding question for links with more than 2 components.
85
Theorem 3
The cokernel of the natural map P2:H2(X;A) § H 2 ( X ; ~ ) is
Tor~(~,A(L)),
and is invariant under 1-equivalence.
The map P2 is onto
if and only if E _|(L) = O. Proof
The first assertion follows from the Cartan-Leray spectral sequence
for the projection p:(X',p-|(*))§ (X,*) which gives rise to an isomorphism ~ A A(L) ~
and to an exact sequence
O § Tor~(=,A(L))
§ =
~ A H2(X;A) + H2(X;~)
~ Tor~(=,A(L))
~ O.
Let M denote the l-adic completion of a A-module M and let T = Tor~(=,A(e)).
Then ~ = ~ and T = ~ since I ~ = I T = O.
Since the
completion of an exact sequence of finitely generated A-modules is exact, = Tor~(=,~(e)) [13 ; page 203~.
The second assertion now follows, as
A(L) is invariant under 1-equivalence, by Theorem IV of Massey.
(See
also Theorem 2).
Now let R be the localization A I, and let A = A(L) I. R/IR = ~ I
= 9, and A/IA = ~
epimorphism $ : R ~ --+ A. So ~ ~ R k e r
|(L) = O.
= Q~, so by Nakayama's lemma there is an
If P2 is onto then Tor~ (9, A) = T I = O,
~ = O and by Nakayama' lemma ~ is an isomorphism.
(Cf. [13; page 84]). E
Then
Hence
~(L) =
A-rank A(L) = R-rank A = ~, so
Conversely, if E _|(L) = O then A(L) is a free ~-module
of rank B, by Theorem I of Massey, so T = ~ = 0 and P2 is onto. //
If L is a 2-component homology boundary link then there is a map f:X + Sly S 1 inducing an epimorphism f,:G § F(2).
Does the inverse image
of the wedge point serve as a singular separating surface for L?
In general
is there a geometrically significant generator for H2(X;A) which projects nicely?
CHAPTER VI
LONGITUDES
AND
PRINCIPALITY
In contrast to the situation discussed in the previous chapter, the t h
Alexander ideal of a ~-component link never vanishes.
Indeed
it is necessarily comaximal with the augmentation ideal I, for evaluating the Jacobian matrix at (1,...,I) gives a presentation for ~ , the abelianization of the link group.
In this chapter we shall be
concerned with links for which E (L) is the first nonzero ideal and in particular when (for such links) this ideal is principal.
We shall
relate the latter condition to the condition that the longitudes lie in G".
These conditions were separately hypothesized as characterizations
of boundary links [174], but we shall give an example to show that they are not sufficient.
We shall give necessary and sufficient conditions
for the Alexander module A(G) to map onto A ~ = A(F(~)) = A(F(~)/F(~)"). In the 2-component case an equivalent condition is that G maps onto F(2)/F(2)" and in this case we can show E2(L ) = (A2(L))(bl(t2) + b2(tl) - 1,(t I - |)be(tl),(t 2 - |)bl(t2) ) for some bl(t2) , b2(tl) in A 2 such that bl(1) = b2(1) = |.
This result,
and the relationship between principality and longitudes were first announced by Crowell and Brown, in the case of 2-component homology boundary links, but no proof has yet been published. Smythe (dated 20 May 1976, E42]),
In a letter to
Crowell stated the following results:
"Let L = 41 U 42 be a 2-component homology boundary link with group G = ~I(S 3 - L). ~LG/G'] Theorem I
Let s,t E G/G' be the classes of the meridians.
= ~ES,s-l,t,t-l]
Then
.
There are polynomials b(s) C ~ [ s ] ,
b(1) = e(1) = I and E2(L) = (A2(L)).I I = ((s-l)b(s),(t-l)c(t),b(s)
where
+ c(t) - I).
c(t) ~ n i t ]
such that
67
Algebraic properties of the ideal I: I.
g.c~d.
2.
I principal~==>I = (1)~===>b(s), c(t) units in ~ [ G / G ' J
3.
I determines b(s), c(t);
4.
E 2 determines I (since it determines A2, and I = (E 2 : (A2))).
Theorem II
((s-l)b(s),(t-l)c(t),b(s)
+ c(t) - I) = I; ;
If L is a boundary link, then b(s) and c(t) are units in
~[G/G' ] . Corollary of Theorems I and II
If L is a boundary link, then
E2(L ) = (A2(t)).
Theorem III ~G/G'
b(s) is a unit (respectively, c(t) is a unit) in
]~---~the longitude
of 4 2 (respectively,
Corollary to Theorems I and III
I principalr
of 4 I) lies in G". >the longitudes of L lie
in G". " His only comment on the proof was that a mixture of algebraic and geometric techniques were used.
In [73]
an argument involving "singular
Seifert surfaces" and Alexander duality in S 3 was used to show that their final corollary holds for any homology boundary link.
The arguments
below, which are of more general applicability, rely instead on equivariant Poincar~ duality in the maximal abelian covering, and the Universal Coefficient spectral sequence.
The Main Theorem If L is a ~-component homology boundary link, so that G maps onto F(~), then A(L) maps onto A ~ = A(F(B)).
In fact it suffices that G maps
onto F(~)/F(~)", the free metabelian group on ~ generators, since the
68
Alexander module depends only on the maximal metabelian group.
quotient of the
(We shall show below that this condition is also necessary
if
=2.) Theorem
1
Let L be a ~-component
link of Alexander
nullity ~, and let B
be the submodule of A(L) generated by the images of the longitudes. (i)
Then
B is pseudozero:
(ii) A(L) ~ Moreover
tA(L) (~ A ~ if and only if E (A(L)/B)
if E (A(L)/B)
maximal pseudozero
is principal
is principal.
then E (L) = (& (L)).Eo(B),
submodule of A(L),
Ann(tA(L)/B)
B is the
= (A (L)/A +l(L)) and
A(L)/B is torsion free as an abelian group. Proof
(i)
nullity
Since ~(L) = ~, every ~-component
v, and so every 2-component
Therefore
all the pairwise
the longitudes G'/G".
sublink of L has Alexander
sublink has Alexander polynomial
0.
linking numbers of L are certainly 0, and so
lie in G'.
Let s
Since each i th longitude
be the image of an i th
longitude in
commutes with an i th meridian,
(I - t i ) . ~ i = 0, and so ~i is in tG'/G" = tA(L).
Now E (L) must contain
some element ~ such that E(~) = l, since e(E (L)) = Z~. Let S={~n]n>,O}. Then A(L) S is projective by the Corollary of Theorem III.9, of ~ annihilates generated by s N >> 0. factor,
tA(L). ' "'" ' ~ '
Therefore
so some power
if B is the submodule of A(L)
Ann B contains -~-(I - ti) and ~N for some i=I
Since c(~) = l these elements
of A have no nontrivial
common
and so (AnnB) ~ = A.
(ii) Let Y be the closed 3-manifold longitudes
of L, and let Y' be its maximal abelian covering
HI(Y; ~) = G/G' (G'/G")/B.
obtained via (0-framed)
(since the longitudes
are in G') and HI(Y;A)
surgery on the space~
Then
= HI(Y'; ~) =
69
Suppose that A(L) ~
tA(L) ~) A ~.
Then since B C tG'/G" = tA(L),
the direct sum splitting of A(L) induces a splitting HI(Y; ~ where T = (tG'/G")/B
is a torsion module and D lies in an exact sequence:
0--+ D--+ A~ --+ A If ~ $ 2 it follows from Schanuel's since it is then of rank at most e~-2D = e ~
= ~
= T (~) D,
and ~ D
s_+ ~
--+ O.
lemma that D is stably free, and
I, it must be free.
= 0 for q # O, ~ - 2 ,
If ~ Z 3, then
while there is an exact
sequence: 0 --+ A --+ A~ --+ e~ By Poincar~ duality H 2 ( Y ; A ) ~ dimension
I.
Universal
Coefficient
E~ q
Therefore
=
are E~ p = e U m
~D
and so has projective
the only nonzero entries
in the E 2 level of the
spectral
eqHp(Y;A) = ~,
and E~ 1 = elH2(Y;A)
--+ 0 .
=
HI(y;A) ~
sequence Ext~(Hp(Y;A),A) ~
E q = eqT ~) ~ D = ~.
HP+q(Y;A)
20 = ~ H 2 ( Y ; A ) (for 0 $ q ~ p + 1 ) , E 2
It will suffice to prove that ~ T
= 0 for
q ~ 2, for then T = tA(L)/B will have a square presentation matrix Suslin's
theorem again) and so E (A(L)/B) = E0(T) will be principal.
If p ~ 6 it is clear from the spectral q > 2.
(using
In general,
sequence
let 6 be as in (i) above.
that eqT = 0 for
Since ~ augments to l,
localizing with respect to powers of ~ does not affect maps between copies of the augmentation module ~. localization
Since ~ annihilates
is an exact functor,
on localizing
tA(L), and since
the above spectral
sequence with respect to powers of ~ all the terms eqT are annihilated. Since H3(Y;A) ~ copies of ~ ought to be.
Ho(Y;A)
= ~,
corresponding
we may conclude that the maps between the
to e ~ ~, eP-2D and elH2(Y;A)
Therefore from the unlocalized
are what they
spectral sequence it follows
70
that eqT = 0 if q > 2, and there is an exact 0 § efT ~ H2(y;A) Now H2(y;A)
~
Hence
is an isomorphism
there
Blanchfield exact
HI(Y;A ) ~
T ~
linking pairing
sequence:
~ e~
D,
§ e2T § 0.
and
e IT ~
e~
=
e~176
=
T (which is essentially
for the cover Y' § Y E ll~)
e~176
the
and a short
sequence 0 --+ D --+ e~176 --+ e2T --+ 0 .
On dualizing
this sequence
isomorphism,
since eqe2T = 0 for q < 2.
is an isomorphism.
it follows
But e~176
from the exact sequences
that e~
Hence e ~
is naturally
defining
Therefore
more easily,
for e2T is then a pseudozero Hence
It now follows
e2T = 0.
isomorphic e~
and hence
that Eo(tA(L))
= (A (L)).Eo(B),
this follows
module with a short free
e2T = 0 and so T has projective
and so E (L) = Eo(tA(L))
to D (as follows
(If ~ $ 2, so D is free,
from Lemma 111.6
is an
a : e~176 ~ e~176176176
D and presenting
e~176 = ~.
resolution).
: e~176176 § e~
dimension
at most
= Eo(tA(L)/B).Eo(B),
since Eo(tA(L)/B ) is principal
and B is pseudozero. The argument are immediate and IV.4
Corollary
in the other direction
consequences
(respectively).
of Theorems
and the remaining
111.9,
lll.10,
assertions
111.4
Remark
//
E _I(L) = 0 and E (L) is principal if and only if A(L) maps
onto A ~ and the longitudes of L lie in G", and in this case G'/G" is torsion free as an abelian group and Ann(tG'/G") = (A (L)/A +l(L)).//
Remarks which
2
]. If E (L) is principal
is contained
in Ann(tA(L))
then A (L) is in E (L) = Eo(tA(L))
by Theorem
III.I
and so the arguments
I.
71
involving 6 in (i) above may be simplified. 2.
If there is an epimorphism n : G * F(~)/F(B)"
since F(~)/F(B)" is residually nilpotent 3.
then kern = G(~),
[145; page 76] .
The conditions in the corollary on A(L) and on the longitudes
of L each imply E _;(L) = 0; independent.
otherwise all four conditions are
For instance the link L 1 U L 2 of Figure V.I has
EI(L 1 U L 2) = 0 and has its longitudes in G", but E2(L) ~
(2-tl,
pseudozero.
I+ t2)(2t I- I, 1 + t 2) and tA(L) = A 2 / ( 2 - t I, 1 +t2)
The link depicted in Figure
is
II.l is a 2-component homology
boundary link for which E2(L) is not principal. 4.
(A late insertion).
An unpublished
theorem of McIsaac
implies that a metabelian group H such that H/H' = ~ free metabelian
[201].
and Webb
and A(H)..N, A ~
is
Hence the above corollary can be restated
entirely in terms of the link group.
A Remarkable Example In this section we give an example of a 2-component ribbon link with trivial components and which has the same Alexander module as a trivial 2-component link, yet which is not even an homology boundary linko it is a counter-example
to Smythe's conjecture,
although it cannot be
distinguished
from the trivial 2-component
invariants.
That it is not an homology boundary link is proven by
showing that G/G
Thus
link by the usual metabelian
is one of the nonfree parafree groups of Baumslag
[ 9 ]-
This link has the additional noteworthy properties
that its longitudes
lie in G", and that G is a split extension of G / G
(and thus is a semi-
direct product).
72
Figure !
Let L = L I U L 2 : 2S 1 -~ S 3 be the link represented by the solid lines in Figure ! and let G be its link group.
Then G has a presentation
{a,b,c,d,e,f,g,x I a-lxa, f-lx-lf, aba-l.c-I b-lx-lfxb.d-lg-ld, cec-logd-lg-l,
b-lfb.d-le-ld,
represent Wirtinger generators associated with the
arcs so labelled in the figure.
The longitudes of L are represented by
fa -I and by c -Iga -I eg -I edb -Ixbd -I x -I .
ga -le-lo
cac-l.gb-lg-I ,
ede-lc -1, xgx-la-I },
where a,b,c,d,e,f,g,x
~,y,8~e,~,e
,
Introduce new generators
and new relators ba-16 -I, ca-Iy -I, da-l~ -I , ea-le-l~ fa-l~ -I , Then the above presentation of G is Tietze-equivalent
{a,x,B,y,8,g,r
to
I x~ x-I ~-I , afla-Iy-I , ~-I x-i ~ ax6~-la-I e-I 6,
6-1#a~-la-l~-1~, EaSg-la-ly-l,
yay-10B-la-le-l,
yasy-16~-la-le -I ,
x@ax-la-I }
and the longitudes are represented by ~ and a-l~-lece-Iya~6-1xflS-Ix -I
73
Clearly 8,~f,~,e,~,8
represent elements of G', and yey-IB~-IB8-1
represents the trivial element of G, so
Be~ -I represents an element of G".
Thus G/G" has a presentation {a,x,~,y,~,~,~,8 I x~x-l~ -I , aBa-l-f-I , x-l~ax~-la-le-i e , ~a~-la-l, Tay-IB-18a-I e-l, ~r
8ax-la-lx, where
[ [ , ],[
[[
, ],[
, ea~-la-l~(-l,
, ]] }
, ] ] denotes the set of all commutators of commutators
in the generators~
It follows that ~Sa.a-l~-18-1s
and so E and hence ~ and 6B-I represent ! in G/G"o longitudes of L lie in G").
represents ! in G/G", (Consequently the
The presentation for G/G" is therefore
equivalent to {a,x,~,y,e I a6a-iy-', ~ay-iB-le-la-le -I , 8ax-la-lx, [[
, ],[
Then aBy-IB-18a-18 -I represents ] in G", and so yeaS-la -I do also.
, ]] } o
and Ba-leae -I
Thus this presentation is equivalent to
{ a,x,e I 8ax-la-lx, [ [ , ],[ { a,x I [ [ , ] , [
, ] ] }9
, ] ] } and finally to Thus G/G" ~
F(2)/F(2)", the free metabelian
group on 2 generators. As L is visibly a ribbon link, the projection of G onto G / G
factors
through H(R), where H(R) is the group with presentation {a,b,c,g,xlaba-IG -I, cac -l.gbg-l,xgx -l.a -I } (so H(R) = G/<< B~ -I, c, ~>>).
This presentation
is equivalent to {a,b,x,tlabab-la-l,x-laxb-lx-la-lx,t.x-la-lxaba-I } and thus to {a,x,tlt.x-la-lxa.t-lata -I } . group G_I,! [ 9 ].
Thus H(R) is isomorphic to the
of Baumslag, which is parafree (so H(R)m = I) but not free
Thus L cannot be an homology boundary link (for otherwise an
epimorphism G * F(2) would induce an isomorphism G/G would then be free).
co
~
F(2) and H(R)
Since the longitudes of L are in G" and since the
74
meridians
a,x map to a generating
counter-example
to Questions
set for F(2)/F(2)"
I and 2 of [174].
link ~ = L U L 3 is an homology boundary Since L is not a boundary if the link of question
(The three component
link whose longitudes
link L cannot be a boundary
1 of [174]
link, it need not be a boundary
is assumed
link.
implies that a slice link with unknotted is a boundary
From the presentation
Thus even
Lambert had earlier constructed
The link is also a counter-example
generated by 2 meridians
link.
lie in G".
to be a homology boundary
an example of this kind, with only 2 components, [110].)
this link is a
but 51 crossings!
to lemma 8 of [63],
components
such that G/G" is
link.
first given for G and H(R) it follows easily
that there is a splitting homomorphism is a semidirect product G ~ G
~
H(R) + G for the projection,
H(R).
G~ = [G, G ] .
to its own commutator
It seems unlikely
subgroup),
although as G
the
link
L interchangeable,
[76~o)
that G~ is perfect
(equal
is the normal closure of
the single element represented by s, G /G ' is a cyclic ~ [ G / G Is
so G
(It may be shown that the
group of the link of Figure V.I is not such a semidirect product Consequently
which
]-module.
that is, is there a homeomorphism
h : S 3 ~ S 3 such that h o L 1 = L 2 and h o L 2 = L 1 ? This example was motivated by McMillan's W : S1 v S1 § S constructed
such that ~I(S 3- imW)
~
example of an embedding
H(R)
by finding a Wirtinger presentation
[127],
and was
of deficiency
2 for H(R)
and then forming a ribbon link whose ribbon group had that presentation.
2-Component Links For links with 2 components, various ways.
the above results
can be strengthened
in
75
Theorem 2
Let L be a 2-component
link.
Then the following are
equivalent (i)
G maps onto F(2)/F(2)";
(ii)
A(L) maps onto A 2 2;
(iii) G'/G" maps onto A 2 . Proof
(i) ~
on G/G".
(ii) Crowell showed in [37]
An epimorphism
that A(L) depends functorially
: G + F(2)/F(2)"
which splits since F(2)/F(2)"
induces a map : G/G" § F(2)/F(2)" 2 Hence A 2 , the
is free metabelian.
Alexander module of a (trivial)
2-component
link with group F(2), is a
direct summand of A(L). (ii)~(iii)
The Crowell exact sequence gives rise to an exact
sequence 0 § (G'/G")/(tG'/G") and so (G'/G")/(tG'/G")
+ A(L)/tA(L)
+ A + Zg § 0
is stably free, by Schanuel's
lena,
and hence
free, since it is of rank I. (iii)--~ (i) ensions
Let H = (G/G")/(tG'/G")o
Then H'/H" ~
by A 2 are classified by H2(ZZ 2 ;A 2) ~
of ~ 2
A 2.
The ext-
2Z, and it is easily
checked that if ] ~ A2 + EX n + ~ 2 is an extension corresponding and that Ex 1 ~
Ex I.
Remark
that H ~ F ( 2 ) / F ( 2 ) "
that "G maps onto F(~)/F(~)" theorem for nonprojective
of extensions
of ~
~
~
~
~2
~
(A2/nA2)
and hence that
//
For ~ > 2, the only difficulty
((G'/G")/(tG'/G"))
1
to n E ~ , then EXn/EX d
It follows
G maps onto F(2)/F(2)".
+
" is the lack of an adequate cancellation
modules ~
is deducing from "A(L) maps onto A~''
(Schanuel's
(F(~)'/F(N)")
by F(~)'/F(~)"
~
len~na shows that ~
and the classifioation
is easy).
See Remark 4 on page 71.
78
Theorem 3
Let L be a 2-component
link with Alexander nullity 2.
Then
the following are equivalent: (i)
E2(L) is principal ;
(ii)
EI(G'/G")
is principal ;
(iii) p.d.A(L) ~ I ; (iv)
p.d.G'/G" ~ I .
Proof
(i) ~ ( i i ) , ( i i i ) , ( i v ) .
A(L)~
tA(L) ~) A ~
If E2(L) is principal then
and tA(L) has a square presentation matrix by
Theorem III.9, whence
(iii) holds, and G'/G" ~
tA(L) ~) A 2
by Theorem 2
so (ii) and (iv) hold. (ii) ~
(i),(iii),(iv).
(iii)~>
(iv).
(iv) ~-~(ii).
This is similar.
This follows from Lemma IV.2. Suppose that p.d.G'/G" $ ].
lie in G", by Theorem IE.10 and Theorem I. obtained by surgery on the longitudes of L.
Then the longitudes of L must Let Y be the closed 3-manifold It
Then HI(Y;A) ~ G ' / G " .
follows as in Theorem I that there is an exact sequence H2(y;A) § e~ and that H2(y;A) ~ rank I, e~
HI(Y;A) ~
A.
) § e2(G'/G '')
by Poincar@ duality.
Therefore since
Since H2(Y;A)
is of
p.d.G'/G" ~ I, G'/G" = H2(Y;A)
maps onto A = A, so G'/G" ~-~ A (~ tG'/G", p.d.tG'/G" ~ I, and (ii) follows from Theorem III.9o// Our next result includes the assertion of Crowell and Brown on the structure of (E2(L):(A2(L)) Theorem 4
for L a 2-component homology boundary link.
Let L be a 2-component link with Alexander nullity 2, and
let B be the submodule of A(L) generated by the longitudes.
Then there
77
are elements bl(t2) , b2(tl) ~ A 2 with bl(1) = b2(1) = I such that B~
(A2/(bl(t2),t I -I)) (~ (A2/(b2(tl),t 2 - 1 ) ) .
AnnB
Hence
= Eo(B) = (bl(t2) + b2(t I) - l,(t I -l)b2(tl),(t 2 - l)bl(t2)),
and B
is torsion free as an abelian group. Proof
Let B 1 and B 2 be the cyclic submodules
and second longitudes
of L respectively.
of dimension 3, and since p.d.G'/G" finite length. have projective Therefore
Therefore
of B generated
by the first
Since A 2 is a regular domain
~ 2, G'/G" contains no submodules
of
the same is true of B 1 and B2, so they each
dimension less than 3, by Remark 2 after Theorem III.ll
in particular B 2 has a finite free resolution b b+l Q 0 --+ A2 --+ A 2 --+ A2 --+ B2 --+ O.
There is also a short exact sequence:
0 --+ A 2 Since t 2
-
I annihilates
t2-1 n --+ A2 -~ A 1 --+ 0 .
B2, it follows that
A2 Tor I (AI,B 2) ~
A 1 ~A2
B2
=
B~.
Then there are exact sequences: 0 --+ ker Q -+ A b+l 1 --+ A 1 1 --+ B 2 --+0 and b 0 --+ A I --+ ker Q -+ B 2 --+ O, where Q is the reduction of Q under the ring homomorphism len~a and Suslin's theorem ker Q considered as a A1-module, the annihilator b2(tl). homologous
~.
By Schanuel's
is a free Al-mOdule , of rank b, and so B2,
has a square presentation matrix.
ideal of B 2 in A 1 is principal,
Since AI(L) = 0 the longitudes
Therefore
generated by some element
of L are in G', that is are null-
in X, and so b2(t l) must augment to a generator +| of Zg, which
may be assumed to be +I. by Theorem IV.4.
Therefore B 2 is torsion free as an abelian group,
Considering
B 2 now as a A2-module,
we conclude that
78
Ann B 2 = (b2(tl),t 2 - I) where b2(1) = I.
Similarly Ann B 1 = (bl(t2),tl-l)
for some bl(t 2) such that bl(1) = I, and B I is torsion free as an abelian group. Let 41 and 4 2 be generators al(tl,t2),a2(tl,t2) Then al(tl,t2).41 (t I -I))
= 0
of B I and B 2 respectively.
E A 2 are such that al(tl,t2).41 = al(tl,t2)ob2(tl).41
(since b2(tl).42
similarly a2(tl,t2)
+ a2(tl,t2).42
= 0o
(since b2(t I) ~ 1 modulo
= 0).
Therefore al(tl,t 2) is Ann BI, and
is in Ann B2, and so B = B 1 (~ B 2 ~
(A2/(b2(tl),t 2 - I)).
Suppose that
In particular,
(A2/(bl(t2),tl-l)(~
B is torsion free as an ahelian group.
Let p = bl(t 2) + b2(tl) - I, q = (t I -l)b2(tl),
r = (t 2 - ;)bl(t2),
s = (t I - l ) ( t 2 - I), t = bl(t2)b2(tl) , b~ = (bl(t 2) - I/t 2 - I) and b~ = ( b 2 ( t l ) - l / t 1 - 1 ) . p = t-s.b~
Then Eo(B ) = (q,r,s,t) = (p,q,r) since
.b~ and t = b 2 ( t 2 ) . P - b ~ . q and s = - s . p + (t 2 - l).q+ (t I - l).r.
Clearly also Eo(B) ~ Ann B = Ann B 1 N Ann B 2. Suppose that a(tl,t 2) is in Ann B.
Then
a(tl,t 2) = m(tl,t2).(t I- I) + n(tl,t2).bl(t2) = m(tl,|).(t I -I) -
+ n(tl,l).(l-b2(tl))
m(tl,l)).(t I -l).b2(tl)
(since it is in Ann BI) + (m(tl,t 2)
+ (t I -|).(m(tl,t2)
- m(tl,l))obl(t2)
- (m(tl,t2) - m(tl,l)), p + (n(tl,t2) - n(tl,l)).b1(t2) Therefore
(invoking the Remainder Theorem to conclude t 2 -I
m(tl,t 2) - m(tl,l) m(tl,l)(t I -I)
and n(tl,t 2) - n(tl,l)
+ n(tl,l), p.
divides
) it follows that
+ n ( t l , l ) . ( l - b 2 ( t l ) ) is also in Ann B, and so equals some
u(tl,t2).(t 2 - I) + v(tl,t2).b2(tl) t 2 = I it follows that
(since it is in Ann B 2 ) .
m(tl,l)o(t I -I) + n ( t l , ] ) . ( 1 - b 2 ( t l ) )
and on setting t I = I, it follows that v(|,l) = 0, so v(tl,l) (by the Remainder Theorem again).
Thus
On setting = V(tl,l).b2(tl), = w(tl).(t I - I)
79
a(tl,t2)
=
(m(tl,t2) - m(t1,1 ) + W(tl)).(t I -l).b2(tl) + (t l-I).(m(tl,t2) - n(tl,])).bl(t2)
Therefore
- m(t!,l).bl(t2)
+ (n(tl,t 2)
+ (n(tl,l) + m(tl,]) - m(tl,t2)), p.
a(tl,t 2) is in (p,q,r) and so Ann B = Eo(B) = (p,q,r)
= (bl(t2) + b2(t I) - 1, Corollary
(t I -|),b2(tl) ,
If G maps onto F(2)/F(2)"
(t 2 - |).b2(t2) ). //
then G'/G" is torsion free as an
abelian group. Proof
For then by Theorem IV.4 and Theorem I (G'/G")/B is torsion free
as an abelian group. //
Meridians
We conclude this chapter with a comment on 2-component boundary links. even
To show that an homology boundary link is not a boundary link, though the ~ th Alexander ideal is principal,
it must be shown that
no set of ~ meridians maps to a set of generators for G / G
= F(~).
There
is an algorithm due to Whitehead for deciding whether a given set of elements of F(~) generates
the group [123; page 166], but here the
possibility of replacing elements by conjugates must also be allowed. If ~ = 2, a theorem of Nielsen leads to a simple answer. Definition w2,...,w ~
The element w I in F(~) is primitive if there are elements in F(~) such that {Wl,...,w~} generates F(~);
equivalently,
if there is an automorphism ~ of F(~) such that ~(w I) = Xl,
where F(~)
is the free group on the letters {xl,...,x }. Theorem
(Nielsen [123; page 169]).
There is at most one conjugacy class
of primitive elements of F(2) with given image in F(2)/F(2)'
=
~2
.
80
Therefore conjugates
if w I and w 2 in F(2) generate F(2) modulo F(2)',
of w I and w 2 generate F(2) if and only if they are each
primitive.
For clearly this is necessary.
primitive.
Then after an automorphism
Suppose w I and w 2 are each
~ of F(2) it may be assumed that
ab ~(Wl) = x I and ~(w2) = x I x 2
modulo F(2)'
modulo F(2)', b must be •
• But the element x ~ x 2
primitive,
and so ~(w 2) = z x ~ x ~ I z -I
(~-l(z))-IWZ(~ -!(z))
for some z.
is clearly Therefore w I and
this result by showing that the link of Figure II.]
is not a boundary link.
boundary
Since w I and w 2 generate
generate F(2).
We shall illustrate
argument,
some
(We thereby avoid appealing
to a Seifert surface
to show that the second Alexander ideal of a 2-component link is principal).
The ribbon group of this link has a
presentation {a,w,x,y,z ] axe -I = y, wyw -I = z, zwz -I = x} which is Tietze-equivalent
to
{a,w,z ]wazwz-la-lw -I = z} and hence to {b,w]~} where b = war. the link is an homology boundary a and w are represented generators
b and w.
cyclically
w-lb2w-lb -I
and w in the free
Since a and bw -2 have the same image in G/G',
to bw -2.
reduced,
link) and the images of the meridians
by the words
since bw -2 is clearly primitive, is conjugate
Thus the ribbon group is free (so
the link can only be a boundary
and
link if a
But the words w-lb2w-lb -I and b w -2 are clearly each
and of distinct
lengths,
and so do not represent
conjugate elements of the free group {b,w[#).
E|23; page 36].
Bachmuth has shown that the analogue of Nielsen's also for the free metabelian
theorem holds
group of rank 2, F(2)/F(2)"
E6].
His
results have been used by Brown to prove that the A -module F(W)'/F(~)" is n o t the direct sum of
two proper submodules
EI7~.
81
Osborne and Zieschang have given a simple procedure for finding a primitive word ~n the coset of Xl m x2 n modulo F(2)' whenever (m, n) = | [208~.
Their formulae apply also in the metabelian case. Suppose finally that L is a 2-component link such that G maps onto
F(2)/F(2)".
If L is a boundary link then there is a pair of meridians
in G which maps to a generating set for F(2)/F(2)". is this the case?
(Remark. 1-1ink
In general when
Is it so if E2(L ) is principal, or conversely?
We should have observed earlier that for a ~-component boundary L,
the ideal
E (L)
is principal.
This follows from the Corollary
to Theorem 1 and the fact that the longitudes lie in
G' c- G'
(page 14) or
more directly from the Mayer-Vietorls sequence of the maximal abelian cover determined by a set of disjoint Selfert surfaces, which gives a square presentation matrix for
tG'/~'.)
CHAPTER Vll SUBLINKS
In this chapter we shall relate the Alexander invariants of a link to those of its sublinks.
For the first Alexander polynomial this was done
by Torres, who used properties of Wirtlnger presentations of link groups to establish two conditions on A I ~891 .
Sato showed that one could derive
Tortes' second condition from the Wang sequence and excision [165]. Traldi has extended the second condition to the higher Alexander ideals [190].
We shall show that Sato's argument applies equally well in this
case.
(Torres' first condition can be deduced from the second condition
and duality if all the linking numbers are nonzero, and then the general case follows by a simple argument due to Fox and Torres ~ ) .
We shall
give some simple consequences of the Torres conditions and state without proof much stronger results recently announced by Traldi ~ 9 1 ,
192].
The Torres conditions (for ~ = I) serve to characterize the first Alexander polynomial of a knot. D673).
(This was done much earlier by Seifert
Bailey and Levine have shown that they characterize the first
Alexander polynomial of a 2-component llnk with linking number 0 and • respectively, while Kidwell has shown that for linking number 3 and under restrictions on the "order" of the link further conditions are necessary ET, 99, 117~.
We shall show that without any such restrictions the
Torres conditions are in general not sufficient in the 2-component case. Our theorem invokes a derivative of the Alexander polynomial that Murasugi had earlier shown was an invariant of certain link homotopies, and so we shall sketch a proof of
Murasugi's result.
83
The Conditions
Let L be a B-component .th th l and j components. be the homomorphism
of Torres and Traldi
link and let s
(Recall s
lj
be the linking number of the
= O for all i).
Let ~: A~ + A _ 1
sending t. to t. for i < B and sending t 1
to I.
Then
the two conditions of Torres may be stated as follows:
(I)
If ~ = 1
, ~
= t2aAl(L)
if
, ~
= (-I)~(
> 1
b.
(2)
~
1 -
Z
for some a
~
~..
;
tibi)Al(L)
modulo
where
(2).
If u = 1
, #(El(L))
= Zg ;
if ~ > ;
, $(EI(L))
= ( K t. l~ _ I) EI(L~) l~i~u i
where L ^ is the sublink obtained by deleting the
th
component of L.
(Note that the first condition does not depend on the choice of first Alexander polynomial= conditions
and that deleting other components
of L leads to
similar to (2)).
The first condition may be restated in the following
slightly
weaker form: (I)'
The principal
ideals
(AI(L)) and (AI(L)) are equal.
We shall prove the following extension of (I)', first obtained by Blanchfield.
(Our argument
is related to his).
84
Theorem ]
(Blanchfield [||]) For each i ~ ], the principal ideals
(Ai(L)) and (Ai(L)) are equal.
Proof
Since A is a factorial domain it will suffice to show that A.(L) i and A.(L)I have the same irreducible factors. Let /&= (p) be a height I prime ideal, generated by an irreducible element p. divides Ai(L ) if and only if it divides Ai(L ). #~
If ~ = ~
, then pa
So we may assume that
, and hence that t.-]l is a unit in the localization A~.
On
localizing the long exact sequence of equivariant homology for the maximal abelian cover of the pair (X, ~ X), and on observing that annihilates H,(~X:A), we conclude that ~ = Hk(X, SX;A ~ H2(X;A~.
for all k.
~
(ti - 1)
~(X;A)~ is isomorphic to
By Poincar4 duality HI(X,~X;A)~
is isomorphic to
The Universal Coefficient spectral sequence then gives an exact
sequence 0 + elMl --> MI~ --+ M2~ --+ 0
Since M2~is torsion free and rank MI~ = rank MI~ = rank M2~ , there is an isomorphism el(tMl~) = e l M l ~
tMl~.
If N is a finitely generated torsion
module over a principal ideal domain, there is an unnatural isomorphism N ~ piN, by the structure theorem for such modules.
The theorem follows. //
We shall use localization in conjunction with duality again in Chapter IX in order to construct an invariant of link concordance.
For knots it is not necessary to localize, as HI(X;A ) = HI(X>~X;A ) and so the Universal Coefficient spectral sequence and duality imply directly that there is an isomorphism elHI(X;A) ~ HI(X;A).
Since HI(X;A)
has a short free resolution with a square presentation matrix, it follows
85
El(L)
= Ei(L)
for all i, if ~ = I.
as condition
(1)' is equivalent
Theorem IV.3.
This is also true for ~ > I, if i=l,
to " El(L)= El(L)
Is it true in general?
" by part
The Steinitz-Fox-Smythe
may be used to show that there are knots
for which HI(X;A)
H I (X;A) and hence which are noninvertible
~I,
( 1 ) ' and (2) t o g e t h e r imply (1).
AI(L)
= u. AI(L)
(2) implies
that
that AI(L)(-I) AI(L)(-I) general
for some unit u = (-I) s
For ( l ) ' b.
Since
is odd (and hence nonzero),
= AI(L)(-I)
is not isomorphic
s
ij
are nonzero,
i s e q u i v a l e n t to
E t. z in A . l~i~ z
e(u) = 1 so that u = tb.
invariants
92].
If ~ = I or if ~ ~ 2 and all the linking numbers conditions
(i) of
AI(L)
If ~ = I,
(I) = •
it then follows
implies
that
= (-I) b AI(L)(-I) , so that b = 2a for some a.
A = u A implies
that ~(A) = ~(u)~(A),
(tl s - I) AI(L I) = tl ~ - I
In
so if ~ = 2 and ~ = 412
~(u)(tls
AI(LI) 4. z~
while
if ~ > 2 (H - I)AI--~.~) = -H~(u)(H
- I)AI(L ^) where
If all the l i n k i n g numbers ~ l i a r e n o n z e r o , (l).
Otherwise,
adjoin a new component
K
o
H =
E t. l~i~ ~
a simple i n d u c t i o n now g i v e s such that ~
oi
= ~(Ko,L i) is
+ nonzero
for I ~ i ~ ~, and let L
be the (~+])-component
link K IL L. o
to conclude AI(L +)(to,...t ~) satisfies c. ( E t i z)AI(L+ ) with o~i~
We may now use the above argument (I), so that AI(L +) = (-I) ~+I
ci ~ ! - o ~ j ~
~ij
modulo
(2). +
On applying we see that
(2) to the link L obtained
by deleting
the component
K
o
ofL
,
to
86
t. io _ I l.
A I(L) =
H l.
H t. Io l
and hence, on cancelling the nonzero factor
sides, - AI(L) = (-I) ~+I
modulo (2).
H l~i~
b. t i IAI(L )
Thus (I) is proven.
'
H 1~i~
- l]A~(n)
t. Io _ I
from both
where b. = c. + ~. ~ I Z l i lo l~j~
ij
(This derivation of (I) from (2) and
duality is due to Fox and Torres [52].
In ~89], Torres had shown that
n.
AI(L) = (-l)~
H l~iz~
t i IAI(L) for some integers ni, but he did not determine
the values of n. modulo (2) there). i
Using methods similar to those of Torres, Traldi extended condition (2) to the higher Alexander ideals.
We shall state his
results in the next theorem, but we shall use "pure algebraic topology" instead, following Sato's proof of (2) [165].
For the remainder of this
section we shall assume that ~ > ], as (2) is clear in the case of a knot. (In Chapter IV we showed that e(E (L)) = ~
for any ~-component link L.
The argument of the following theorem may be applied also in the knot case, with slight changes).
Theorem 2 (i) (ii)
(Traldl [19~)
~(EI(L)) = ( H -
Let L be a p-component link, with ~ > I.
Then
l) EI(L~) ;
Ek_I(L ~) + ( H -
I) Ek(L ~) C ~(Ek(L)) ~ Ek_I(L~) + I _iEk(l ~)
for each k ~ 2. (Here as before ~ = the last component of L).
t. i~ and L2 is the sublink obtained by deleting
87
Proof
Let X be the exterior of L and Y the exterior of L^, and choose
a basepoint * in X.
Let ~ be the covering space of X induced by the
maximal abelian cover Py : Y' + Y of Y.
We shall let H,(X;A _ I) denote
the eqnivariant homology of X, and similarly for the pairs (X,
py-I (*))
and (Y', X).
The cover r:(X',p-l(*)) + (~,py-l(,)) induced by a map from X to S I.
is infinite cyclic, and so is
The Cartan-Leray
spectral sequence for r
is just the Wang sequence for the fibration X' + X + S I, and gives an exact sequence A(L) ~
A(L) ---+ HI(X,*;A _ I) ~
O
Hence we shall abbreviate HI(X,*;A _ I) as ~A(L).
The long exact sequence of
equivariant homology for the triple (y,,~,py-](,))
H2(Y;Au_ I) ---+ H2(Y,X;A _i) ~
~A(L) ---+ A(L^)
gives an exact sequence.
~ HI(Y,X;A _l).
By excision HI(Y,X;A _i) = O and H2(Y,X;A _i) is isomorphic to A
i/(~-I).
Suppose first that H2(Y;A _ I) = O, in other words that EI(L ~) # 0. Then A(L~) has a short free resolution, and there is a short exact sequence O ---+ A _l/(~-l) ---+ ~A(L) ----+ A(L~) ---+ O. Since ~(EI(L)) = EI($A(L)) , (i) now follows from Lemma 111.6.
If
H2(Y;A _l) ~ 0 (i) is trivially true, as both sides of the equation are then O.
In general there is an exact sequence 0 ~
A _|/J ~
SA(L) ~
where J is an ideal containing
~-l.
A(L^)~ ~
0
Hence if P is a presentation matrix
88
for A(L0), ~A(L) has a presentation matrix of the form
j,
where K*
and J* are column vectors and the entries of J* generate the ideal J. Hence ~(Ek(L)) = Ek(~A(L)) contains Eo(A _I/J)Ek(LQ) and so Ek_I(L 0) + (~-l)Ek(L ~) C ~(Ek(L)). (K,J)Ek(L0).
tl j,
Similarly ~(Ek(L)) C Ek_ I (L0) +
Since Z~ ~ ~A(L) = ZZu and ZZ ~
is in the span of the columns of
and EI(A _i/J)Ek_ ](L 0)
A(L~) = ZZ~-I,
[0I
, modulo I . 1 .
the column
Hence
~(Ek(L)) C-Ek_I(L ~) + I _IEk(L~). //
Traldi gave a number of examples to show that this theorem is best possible,
in the sense that either both inclusions could be strict, or one
or both could be equalities.
Some Consequences of Torres' Conditions
The Torres conditions enable us to argue inductively about the first Alexander polynomial of a link from those of its sublinks.
The
following theorem is an example.
Theorem 3
If L is a ~-component link with ~ ~ 2, then AI(L) is in I ~-2
Proof
It will suffice to show that all the partial derivatives of AI(L)
of order less than ~ " 2 vanish at t I = ... = t
= I.
There is nothing to prove in the 2-component case. holds for all links with fewer than ~ components. D
: f ~--+ ~l~If/~tl~l...~t
~
of order l~I =
We shall induct on ~.
Suppose the result No partial derivation
E.~ ~. less than ~ can
89
involve differentiation with respect to all ~ variables t.. i D
is such a partial derivation with ~
= O.
Suppose that
The Torres conditions imply
that there is some g in A~ such that AI(L ) = (~ - I)AI(Lfl) + ( t
- I)g
(where ~ and L~ are as above). Therefore D AI(L) =
Z
(~!/(a-B)!BI)DB(H-I).D _BAI(L ~) + (t -I)D g where the summation
is taken over multi-indices ~ = (BI,...,B) BI =
H
B.I.
e(D AI(L) ) =
such that Bi ~ ~'I for all i, and
Hence
Z (~!/(ez-B)!B!)( ~ (s 0<8~ l~i<~
!/(s
-~i)!8i!))e(D _8AI(L0) )
If i=i < ~ - 2 and 0 < f~ ~ ~, then Is - Bi < (~ - I) - 2, so all the terms c(D _BAI(L~)) are O, by the inductive assumption, and therefore ~ D
(AI(L)) = O.
Since a similar argument works for the other partial derivatives of order less than ~ - 2, the theorem is proven.
//
If L is obtained from the 2-component link with abelian link group by replacing the second component by ~ - I parallel pairwise unlinked copies of itself, then AI(L ) = (t I - I) ~-2 and so this result is in general best possible.
A similar argument shows that if for every (~-l)-component sublink
LAl of L each partial derivative of AI(L~) of order less than ~ - 2 (respectively, -I) vanishes at t I = ... = t
= I, then all the partial derivatives of
AI(L) of order less than ~ - I (respectively, ~) vanish there. if ~ ~ 2, AI(L) is in Im if and only if El(L) ~ Im+l).
(Note that
If all the linking
numbers ~U are O, the Torres conditions and the Remainder Theorem imply that ( t
- I) divides AI(L ).
Therefore if s
= 0 for all i and j, the
90
Alexander polynomial is divisible by
~
(t. - l) and so is in I ~.
In his thesis Traldi used a similar argument by induction to show that if k < p then Ek(L) ~ I~-k + (ti that Ek(L) ~ I~-k.
I) for each ! ~ i ~ ~, and hence
He has since announced the following stronger result: 4.
"Let M = ~ ~[mij] be the ~ • ~ A -matrix with entries m.. = l -
and, for i # j, m.. = t. lj _ I. lj 1 E Ek+i(L).12i + ~ ( P ~ ) O~i~-k
=
~
9
t. lj
Then for O < k <
I Ek+i(M)-12i + 12(~-k). O~i~u-k
In particular
if 4.. = 0 for all i, j then El(L) ~ 1 2(~-2) and hence AI(L) is in 12~-3"'' lj (This is a paraphrase of part of his announcement of the paper [192]).
Theorem 3 says nothing of interest when ~ = 2. conditions imply that e(Al(L)) = • ~12 in this case.
However the Torres This is also a
consequence of Milnor's theorem as the following proof of another result announced by Traldi shows.
Theorem 4
(Traldi ~91])
Let L be a ~-component link, with ~ ~ 2.
--Let d.l = h.c.f. { ~ij I I ~ j ~ } =
(dl(t
Proof
I -
I) .... d
(t
Let B = G'/G".
IB ~ 12A(L).
-
I))
+
for each ! ~ i ~ ~.
Then E _|(L) + 12
12 .
Since ~ ~ A ( L )
= ~
~)I, B C I.A(L) and so
Therefore A(L)/12A(L) depends only on G/G3, since B/IB =
G'/G3G" = G2/G 3. { x.l
By Milnor's Theorem G/G 3 has a presentation of the form I ~ i ~ ~ [ [xi,~i (3)] = 1
, L
, E
, i] .}
91
where ~.(3) represents an i.th longitude modulo G 3. We may assume i (3) ~il ~i~ hi = x ...x w i for some w.l representing an element of G' = G 2. Let ~.j : F(~) § A be the Fox free derivative such that ~j(xj) = l and ~j(x k) = 0 if j r k. {~j[xi,~i(3)]
Then E _1(G/G3) is generated (modulo 12) by
I 1 ~ i, j ~ ~}.
Since ~.[x i, ~ i ( 3 ~ J
if i # j and ~j[xj, Ej(3)] = [ K t ~ j k ~l~k~ k
I]
= (t i - l)~ij .
~ t l~k~ k
~ik
which is congruent to
E ~jk(tk - I) modulo 12, and since E _ I ( G / G 3 ) l~k~u
+ 12 = E l ( L )
+ 12, the
theorem follows. //
The assertion of this theorem is equivalent to the case k = ~ - I in the above announcement of Traldi, who has advised us that he used the Milnor presentation of G/G
n
also, to obtain his more general results.
has also announced that E _I(L) ~ J + D, where J = E _ 1 (.l ~~_ z~
Traldi
A/(t.-1)) z
is the ideal generated by the products (t i - l)(t. - I) with 1 ~ i # j ~ ~, J dk and where D is the ideal gefierated by the elements tk - I, for 1 ~ k ~ ~. (Notice that D + 12 = (dl(t I - I), ..., d (t
- I)) + I2).
This assertion
implies that of Theorem 4 also.
Insufficiency of the Torres Conditions
Seifert showed that an integral Laurent polynomial ~ in A I which was symmetric of even degree (~ = t 2m 6 for some m) and such that E(~) = • | was the first Alexander polynomial of some knot ~67].
His method was to
embed a punctured surface in S 3 with prescribed self-linking characteristics. Using surgery on the knot complement, Levine showed that there was a knot
92
for which G'/G"~v Al/(6), and by taking connected
sums of knots with G'/G"
a cyclic Al-module he obtained a similar characterization of all Alexander
polynomials
The conditions when U = I.
of a knot
for the family
~ 153 .
~ = t2m6 and ~ )
= •
are just the Tortes conditions
The obvious question is whether the Torres conditions
also when ~ > I. generate Alexander
(See also
DO;
polynomials
Problem 23).
suffice
Levine has shown how to
for links with prescribed
linking matrix
from a given one by surgery and hence has shown that if ~ = 2 and the linking number s is • characterized
the Torres conditions
suffice
the Al-module G'/G" for a 2-component
form of a presentation matrix, also if s = O [73 .
D I~.
Bailey
link in terms of the
and hence has verified
that they suffice
(We shall state Bailey's Theorem below).
shown that for u = 2 and s = 3, and under restrictions of the link, the Torres conditions
are insufficient
has
Kidwell has
on the "order"
~9~.
He has also
shown that if ~ ~ 3 and all the linking numbers are O, then the reduced Alexander polynomial ~00~.
AI(L)(t , ..., t) must be highly divisible by t - l
(See the next Chapter).
However
if ~ > 2, the Torres conditions
do not always suffice to deduce the linking numbers,
and so this result
need not imply that they are insufficient.
In this section we shall assume that ~ = 2 and shall show that even then the Torres conditions determine
the linking number
are in general not sufficient.
(up to sign) when ~ = 2, and as Bailey and
Levine have settled the cases s = O or • Our argument See also
we shall assume that s > I.
shall rely on the following theorem of Bailey
14/4]:
As they
~;
page 32.
93
"Theorem
A A2-module is a link module if it has a presentation matrix of
the form
I - (xy)~/l-xy 8(x-;
where ~(x,y)
,Y
-(l-x)(l-y) (I - (xy)s
-l~tr )
/A(x,y)
is a square matrix,
B(x,y) is a row matrix, both with entries
in A 2, satisfying ~(x,y) =/A(x -I Furthermore ~(x,l)(respectively, first (respectively,
- xy)8(x,y)
y-l)tr and~(l,|) $(l,y))
= diag(•
.,•
is a presentation matrix for the
second) component of the link and ~ is the linking
number of the two components."
The entry in the (l,l)-position of the above matrix is the first Alexander polynomial of the link which bounds an annulus embedded in S 3 with unknotted core and ~ full twists.
Bailey proved his theorem by
observing that any 2-component link with linking number ~ could be obtained by surgery on this link, and by using Alexander duality to compute presentation matrix for the link module. A2-module G'/G".
a
(Here the link module is the
We shall follow Bailey in using x and y instead of t I
and t 2 for our Laurent polynomial variables).
One corollary of Bailey's theorem is that the Alexander polynomial of a 2-component
link has the form
&(x,y) = (I - (xy)~/1 -xy).A(x,y) with A(x,y) = A(x-l,y-l),
- (1 - x ) ( ;
-y)(l
- (xy)~-|/; -xy).B(x,y)
B(x,y) = B(x-l,y -I) and A(x,l) and A(l,y) knot
94
polynomials. B(x,y)
= det
[o :}
(For instance we m a y take A(x,y) ~tr
.)
= det A(x,y),
He showed moreover
that a polynomial
this form if and only if it satisfies both Torres conditions. polynomial
satisfying
the Torres conditions,
its expression
in A 2 has
Given a
in the above
form is not unique.
However
there is some C ~
such that A - A' = (I - x)(l - y)(l - (xy)s
and B - B ' passing and
A2
= (I - (xy)s to a quotient
(I - (xy)s
containing
- xy).C.
We may therefore
ring in which each of
- xy) are mapped
these two elements,
since it is prime and since together generate so ~
if A, B and A', B' both give rise to A, then
to 0.
~.
domain,
unity~ and to consider to ~ and y to I.
=
A2/~
- xy) and
polynomial then ~
Notice
f
then ~d(X)
~ --+
x --+ x
~ = -I
(I - x s
- xy) x)
E
for some d > !
must contain
: A 2 ----+
that the involution
- xy)
! - x or I - y,
(I - (xy)~/l
~ (y) for e
ring to be an integral
to fix a primitive
the h o m o m o r p h i s m
gives rise to complex conjugation R
- y)(l - xy)~-I/l
Thus if we wish our quotient
it is no loss of generality
the ambiguity by
is a proper prime ideal of A 2
If 1 - y E ~
(Similarly if I - x ~ ~ , ~.)
If ~
(I - (xy)~-I/l
the unit ideal.
some e > ] dividing
(I - x)(l
then it must contain either
must contain the d-cyelotomic
dividing
remove
- xy).C
d th root of
R = ~ E ~ ] mapping x -I
, y --+ y
-I
on the quotient
of A 2 ring
.
The images f(A) and f(B) of A and B in R are well defined and so we m a y ask how they may be determined projection
of A 2 onto A I = A2/(I
the knot polynomial Taylor expansion
from A.
The map f factors
through the
- y), and we see that f(A) is the value of
(x - l)(x s - l)-IA(x,l)
of A, A, B etc. about
at x = ~.
On considering
the
(x,y) = (I,~) we see that f(A) is
95
-I also equal to ~ .~(~ - l).~A/~x(~,l), and that f(B) = B(~,I) = ~.(~ - l) -l.(~A/~y(~,l) - ~.~A/3x(~,l)).
It is easily checked
that f(A) and f(B) are real, using the Torres conditions and the fact that complex conjugation is induced by the involution of fi2.
Since the coefficients
of A are rational, whether f(A) = O does not depend on the choice of the primitive d th root ~.
We shall now state and prove our main result
Theorem 5
Let L be a 2-component link with linking number ~ > 1, and with
first Alexander polynomial A(x,y). is (up to a
If the knot polynomial
(x-I/x ~ -l)A(x,l)
unit) the d-cyclotomic polynomial ~d(X) for some d > I dividing
6, and if ~ is a primitive d th root of unity, then the 2Z [~]-ideal generated by ~(~ - l)-l.~A/~y(~,l) is of the form JJ for some ideal J.
Proof
By Bailey's theorem there are square matrices
/A and ]B =
~tr
/A
which are Hermitean with respect to the involution of fi2 and such that A ffi (I - (xy)~/l - x y ) . d e t
/A-
(I - x ) ( l
-y)(l
- (xy)~-I/l - x y ) . d e t
~.
If
~d(X) divides (x - l)(x ~ -l)-IA(x,l) = det /A(x,l), then det f(/A) = f(det /A) = O.
Suppose first that R = ~ ~
is a Euclidean domain.
one of the rows of f(/A) to 0 by elementary row operations.
Then we may reduce Since any elementary
R-matrix may be lifted to an elementary fi2-matrix, we may thus find a fi2-matrix with determinant 1 such that f ( ~ ~ ~ tr) has first row and column O.
(We
perform the conjugate column operations also so as to preserve the Hermitean character of the matrices.) and ~
~tr has the form
Therefore if Q = I ~
then det ~
= det(~
~tr)
96
O
~I
~1
a~d(X)
-tr
+ b ( y - I)
~d(x-l)~tr + (y-I
~d(X)~ + (y - l ) v
1)~
r
for some square matrix r = ~tr, row matrices y, ~ and ~, and elements a = a, h = b, B 1 of A 2.
Then f(det ~)
= f(det(~ 9
~-tr)) = _ f(Bl)f(Bl)f(de t ~),
and f(~d(X)-|det~ (x:,|))= f(~d(X)-Idet(~(x,l)~tr)) Therefore if det~(x,l) = ~d(X), f(det r
= f(a).f(det r
is a (real) unit and so the ideal
generated by ~.(~ - l)-l.~A/~y(~,l) = f(det ~)
equals J], where J is the
ideal generated by f(Bl)"
The above argument is directly applicable for only finitely many d [124], but the general case may be recovered by localization, since R is a Dedekind domain. or R ~
define
Let ~
be a prime ideal of R, and for each ideal I of R
V~ (I) by the equation I~
= IR~Z = ~ V ~
of the previous paragraph (with A2,f-l(~ ) and R ~ shows that (f(det ]B)~ ~ =r
in place of A 2 and R)
= (f(b~)f(bv~)) for some be%, and hence that if
, V ~ ((f(det ]B))) = 2Ve~((f(h~))) = 2Wc~ say.
V~f((f(det IB))) = $ = { ~ #~
(I)~ . The argument
If ~ # ~ ,
then
V~((f(det IB) = Z~ say, since f(det ]B) is real.
Let
I Z~ > O}, and let r C S contain exactly one representative of
each complex conjugate pair.
V~((f(det ]B))) = V~(J~)
Let J = ( ~ W ~ =~
).( ~ ~ Z r ) . ~T
Then
for all primes ~> of R, so we may conclude that
(f(det IB)) = JJ F168; page 23].
This proves the theorem. //
97
The hypothesis
of the theorem is vacuous unless d is divisible by
at least 2 primes, for ~d(1) must divide a knot polynomial, be
•
Therefore
the first case to look at corresponds
then ~6(x) = x 2 - x + I. D(x,y)
=
Consider
and so must to ~ = 6, and
the polynomial
(I - (xy)6/l -xy).(x - 1 + x -l) -(I -x)(l -y)(l - (xy) 5/l -xy).2.
Then it is easily verified that if m is a primitive
that D satisfies
6 th root of unity,
the conditions
of Torres, but
then m.(~-l)-1~D/~y(~,
I) = 2,
and the 2ZEal -ideal generated by 2 is clearly not of the form J]. cannot be the first Alexander polynomial that in this case the ring 7zE~0] (This example,
and subsequently
question in Bailey's
thesis
of any 2-component
Thus D
link.
is actually a~ Euclidean domain
Notice [124]
the above theorem, was suggested by the
[~7; page 69] on whether
there were any
matrices /A, ]B as above such that (det/A, det ]B) = (x-I + x -1, 2). argument of the theorem extends readily to give a necessary a pair of elements
(a,b)~
A22
prime ideal containing
condition for
to be of the form (det/~, det ]B).
E A 2 is a simple prime factor of a such that ~ such that
~ = ~
Suppose
(~) = (~), and that ~
and R = A 2 / ~
c which divides
is a
is Dedekind.
Then the ideal generated by the image of b in R must be of the form for some ideal J and some element
The
cJ~
the image of ~-la ).
98
Murasugi's
Theorem
Although the above theorem follows almost inevitably from Bailey's Theorem, its meaning is still rather obscure.
The cyclotomic polynomials
surely suggest that the homology of a d-fold cyclic cover of the link exterior is involved. not at all clear.
The role of the partial derivative ~A/~y(~,|) is
In this section we shall sketch a proof of a theorem
of Murasugi which shows that the ideal generated by this derivative in ~ ~ ] is invariant under homotopy of the second component of the link.
We are
grateful to Murasugi for sending us an outline in English of his theorem, which has only been published in Japanese [142a,b].
Theorem
(Murasugi)
Let L + and L- be ~-component links which share the same
(~-l)-component sublink K obtained by deleting the t h that L +
is homotopic to L-
in S 3 - K.
polynomials of e + and e - respectively. generate the same ideal in A Proof
components, and such
Let A+ and A- be the first Alexander
~ ~--4 §
Then ~t it =i
and
~
~
A-
t =I
I/(AI(K)).
As the argument is no different in the general case, we shall assume
that ~ = 2 and write x,y for tl,t 2 respectively.
It will suffice to assume
also that L- is obtained from L + by changing one overcrossing of L+ to an underorossing.
Thus we may depict the two links as in Figure
I.
Murasugi's
idea is to compare A + and 4- with the first Alexander polynomial A ~ of the (~+l)-component link L ~ depicted in the figure.
He deduces the theorem on
applying the second TorTes condition and the following lemma.
99
+ L-
L
L~
Figure
Lena
l
For suitable choices of the Alexander
~A+/~y(x,1) + ~ A - / ~ y ( x , l )
=
This lemma is a straightforward
A~
computation,
[40, 139, 189] that the determinant Wirtinger presentation
polynomials,
obtained
based on the fact proven in from the Jacobian matrix of a
of a link group by deleting a column corresponding
.th to an i meridian and any one row is (t i - I) times the first Alexander polynomial
of the link.
The
the first Conway identity
V
+
--
lemma may also be proven by differentiating
E32, 86J:
V
-
=
(y~ - y- 89
o
.
Neither approach is in the spirit of the rest of these notes.
As our Theorem 5
depends on Bailey's use of surgery to change the crossings of components a link with themselves,
in other words to carry out a link homotopy,
expect a deeper connection between it and Murasugi's Theorem, we should seek a proof of the latter theorem via surgery.
of
we might
and in return
CHAPTER VIII
REDUCED
ALEXANDER
IDEALS
The methods of the above chapters may also be applied to the homology of other covering spaces of link complements, and in particular to the infinite cyclic cover determined by the total linking number homomorphism.
For knots this is the maximal abelian cover, and may be
constructed by splitting along a Seifert surface~
As this technique
works for any link, the total linking number cover has been studied extensively.
(See for instance E84,
I00, 138, 172, 182]).
It is of
particular interest when the link complement fibres over the circle; such is the case for the links associated with algebraic singularities E133] , and here the reduced Alexander polynomial is the characteristic polynomial of the monodromy. In this chapter weshall give "coordinate free" proofs of results of Hosokawa and Kidwell on the divisibility of the reduced Alexander polynomial of a link.
The conditions of Torres and Traldi are used to
show that the Hosokawa polynomialv (L) is symmetric of even degree, and to evaluate the integer IV(L)(1)I.
We consider links for which the
rank of the reduced Alexander module is maximal, and prove once again that the link of Figure V.I is not an homology boundary linko
Next we
define fibred links, and show that the reduced Alexander polynomial of a fibred l-link is nonzero, while the only fibred links in higher dimensions are fibred knots.
In this section we list without proof
some of the properties of the monodromy of algebraic links.
We
conclude the chapter with a brief summary of some results on the branched coverings of a link.
101
The Total Linking Number Cover Let L be a b-component group G, and let r : G § ~ meridian
to I.
~Et,t-l]
be the unique homomorphism
This determines
which corresponds each variable
n-link with exterior X, pointed by *, and
a homomorphism
to the projection ~ : A
t i to t.
(Throughout
~ A
sending each
from ~ E G / G ' ~ =
~ E t , t -I]
this chapter A shall denote
The total linking number cover of X is the cover
q : X T + X determined
by ker T.
It is readily verified
that a loop in X lifts to a loop in X T if
and only if the sum of its linking numbers with various
cover".
sending
only.)
Definition
components
to ~ E ~
of L in S n+2 is 0; The Cartan-Leray
whence the name "total linking number
spectral
sequence for the cover q reduces to
the Wang sequence of the fibration X T § X - ~ unique homotopy
(oriented)
S ! (where ~ represents
the
class of maps inducing T), and we shall invoke it as
the "Wang sequence for q". The total linking number cover of an n-link is considered by Sumners in E182],
where he relates
the ~ -
and A-module
structures
the homology of X T, extending results of Crowell for knots gives necessary and sufficient to fibre over the circle. rational homology of X knots
ELI6],
conditions
for an n-knot
Shinohara and Sumners
as a QA-module,
C172~
E38] , and
with group study the
extending results of Levine for
and deduce criteria for the link to be splittable.
shall however not consider the next three sections
on
the high dimensional
shall assume that n = I.
We
case in detail, and in
102
The Hosokawa Polynomial If L is a l-link the equivariant
chain complex of (XT,q-I(*))
is
chain homotopy equivalent to one of the form 0--+ A a ~(d) Aa+l --+ 0--+ 0 where d : h a --+ A a+l
is the boundary map for the corresponding
complex for (X',p-l(*)).
Thus HI(X,*;A)
which we shall abbreviate as ~A(L). Alexander module of L).
= HI(XT,q-I(*);~)
= h @A
A(L),
(This may be called the reduced
Since Ho(*;A) ~
II ~
A, and HI(*,A) = 0,
there is an exact sequence 0 --+ HI(X;A) --+ ~A(L) --+ h --+ 0 so ~A(L) ~
A (~) HI(X;A).
for all i.
Hence Ei(HI(X;A))
In particular Eo(HI(X;A))
= Ei+I(~A(L))
= ~(EI(L))
= ~(Ei+l(L))
is principal, by
Theorem IV.3. Definition
The reduced Alexander polynomial of L is Ared(L)(t)
By Theorem VII.3
=
~(AI(L))
=
Al(L)(t ..... t).
AI(L) is in I ~-2 ' so ~(AI(L))
(~(I)) ~-2 = (t-l) ~-2 and so ~(EI(L)) ~ (t-l) ~-l again, if ~ > l). Definition
is in
(using Theorem IV.3
Therefore the following definition is possible.
The Hosokawa polynomial
of L is the generator V(L)(t) of
the principal ideal (t- l) I-~EI(~A(L))
which satisfies
?(L)(t) = Al(n)(t ) if ~ = l and V(L)(t) = Ared(L)(t)/(t-l)~-2
if ~ > I.
103
Of course this definition depends on a choice of representatives for the first Alexander polynomial, but the ambiguity shall be quite harmless here. Are d and V.
We shall usually abbreviate Ared(L)(t) and V(L)(t) by
Note that bre d # 0 if and only if H2(X;A) = H2(XT;~)
is O.
Hosokawa proved that Are d was divisible by (t - l) u-2 by computing linking numbers of cycles on a Seifert surface
~4].
(We shall give
yet another proof shortly by means of the Wang sequence for q).
The
example following Theorem lrll.3 shows that in general Are d need not be contained in (t - I) ~-| , in other words that e(V) need not be O. Hosokawa showed that V is syn=netric of even degree (~ = t2mv
for
some m), and that any such symmetric polynomial was the H0sokawa polynomial of a ~-component
link, for each ~ > I.
(Thus the ambiguity in the
definition of ? may be reduced to one of sign, by replacing V by tmv, so that tmv = tmv). Furthermore he computed the absolute value of the integer e(V) as a determinant
in the linking numbers of L.
If U = I, the Hosokawa polynomial
is the first Alexander polynomial
of the knot, and it was shown in Chapter VII to be symmetric of even degree and to augment to •
Seifert showed that any such polynomial
is the Alexander polynomial of a knot
~67].
104
If ~ > I, we may derive the symmetry conditions from the first Torres conditions, that AI(L) = (-I)~ I
]~--t~i
I AI(L)where
l ~ i ~
b. ~ I -
~
(~_ |)~-2~
= (_l)~ tZbi (t- I) ~-2 V , so V = tb V where
b
=
2-~
~.. modulo (2).
+
I
b i -=
! .
For then
~
~ij
modulo (2)
is even.
I .< i,j .< ~
Following Traldi, we may deduce the value of IE(V)I from the equality of his announcement
E192~.
We shall first prove a lemma needed below,
which shall also provide another proof that ( t - I ) ~-2
Lemma ;
divides AI(L).
Let W be a finite cell complex with HI(W; ~) ~
p : W~ + W be a connected infinite cyclic cover.
~
, and let
Let H = HI(W~; ~)
considered as a A-module via the action of a fixed generator for the covering group.
Then E.(H)
Proof
C
(t - I) ~-l-i
for
i < ~ .
It will suffice to show for each i < ~ that the ideal generated
by ( t - I ) B-l-i of Ei(H).
in QA = QEt,t-l]
contains that generated by the image
The latter ideal is just E i ( H Q Q ) ,
and H ( ~ Q
is the first
homology of W ~ with rational coefficients, considered as a QA-module. Since QA is a principal ideal domain, we may write H(~)Q =
where 0s+ [ divides e s in QA for I ~ s < r.
Q (QA/(es)) i.<s,
The Wang sequence (with
105
rational coefficients) for the cover p t-I 0 ... --+ H (~W~;Q) ---+ HI(W~;Q) --+ HI(W;Q) --+ Q -+ Q --~ Q then shows that H(~) Q/(t- I ) H ~ Q
Q~-I.
Hence the first ~ - 1
=
~) (Q/es(1).Q) is isomorphic to l.<s.
of the numbers es(1) must be 0 and so
11
i+ .<s~
(As remarked in Chapter VII it is in fact true that
Ej (e) C I ~-J for j ~
On applying this lemma to Traldi's equality
in the case k = l , we see that ~(EI(L)) =- ~(EI(M)) modulo (t-l)~o fact EI(L ) E EI(M ) modulo I ~(EI(L))
=
(t- l)~-l(v)
~(mii)-
[-
[
). =
Now
(t- I)Z-I(e(V))
%ij]It)~ I])
I.<j.<~
modulo (t-l) 2 if i # j,
so
(In
modulo (t- I)~,
while
modulo (t-|) 2 and ~(mij) - s
~(EI(M)) = EI(N).(t- I)~-| modulo (t- I)~
where N = [nij ] is the ~ x~ ~-matrix with nil = -
[
s
and
] ~ J ~
nij = s
if i # j.
Hence s(V) and El(N) generate the same ~-ideals, in
other words le(V) I = [El(N) I.
(Note that since the sum of all the rows
and the sum of all the columns of N are each O, all the (~- l) • (~- |) minors of N have the same absolute value.) If all the linking numbers are nonzero this last result may also be established via the Torres conditions, starting from e(V)
=
lim ~ed / (t-I) ~-2 t+ I
-
I e(d~-2bred/dt~-2) (~ s 2)!
=
[~[ 7~-2 6~(AI(L))
106
where
1 le~[ a1 6 f = - - e(a f/at I
index ~ = (al,...,~). computed
... ~t
c~>
)
for any f in A and any multi-
The terms 6 (AI(L)) with
inductively using the Torres conditions,
terms in the expansion of a principal
la] = > - 2
may be
and compared with the
(m- I) x (>- I) minor of N.
The
assumption on the linking number is needed to keep track of the ambiguity in sign of the Alexander polynomials In E89],
of the sublinks.
Kawauchi has defined Hosokawa polynomials
infinite cyclic coverings,
for more general
and has used an extension of Milnor duality to
prove that these polynomials
are of even degree and are symmetric up to
sign.
Kidwell's
Theorem
In this section we shall show that the divisibility powers of t - I can be more than doubled The following
Theorem 2
(Kidwell
spanning surface for
EI00~)
Let e be a D-component
of L have mutual linking number O.
Are d = (t-l)2~-3f(t)
0-framed
from a connected
of Hosokawa's
We shall assume that Are d # 0.
two components
Proof
if all the linking numbers are 0.
theorem was proven by Kidwell by a refinement
approach, using a matrix derived the link.
of Are d by
link such that any
Then
for some f(t) in A such that f(]) = O, if__ ~ is even.
Let W = X U ~(D 2 • S I) be the closed 3-manifold surgery on the link L.
Since all the linking numbers of L
are 0, the inclusion map HI(X; ~) + HI(W; ~) the total linking number cover extends By Lemma I, Eo(HI(WT;Q))
obtained by
is contained
is an isomorphism,
and so
to an infinite cyclic cover r : W T + W . in ( t - l ) ~-I.
sequence of the pair (WT,X r) gives an exact sequence
The long exact
107
0 --+ H2(WT;Q) --+ H2(Wr,WT :Q) --+ HI(XT :Q) -+ HI(WT :Q) -+ 0 (in which the first map is injective since Are d # 0 implies that H2(XT,Q) = H2(XT; ~) Q so
Q = 0).
H2(W~;Q) = H2(W;QA) ~
Therefore HI(WT;Q)
HI(w;QA) ~
EXtQA(Ho(W;QA),QA)
by duality and Universal Coefficients. isomorphic to H2(r-I(D2•215 Eo(HI(Wr;Q)).( t_ 1)~
=
is a QA-torsion module,
By excision,
SI);Q) ~
~
Q = QA/(t-|)
the second term is
(QA/(t-I)~).
Eo(HI(XT;Q)).(t_ ]) = (Area( t_ ])2),
(t-l) 2~-3 = (t-l) u-I+~-2
Therefore so
divides Are d in QA, and therefore also in A.
(Notice also that we have shown Eo(HI(WT;Q))
= (V).)
Let f(t) = Ared/(t-l) 2~-3 = V / ( t - | ) ~-I. some m, f(t) = (-])~-] t 2m+~-| f(t)
Since V = t2mv
and so f(1) = (-l)~-If([).
for Therefore
if U is even, f(]) = O. // Kidwell proved that the integer (~
-
1) • (~ -1)
If(1) I is the determinant of a
skew symmetric matrix derived from Hosokawa's matrix, and
so is 0 if ~ is even and is a perfect square if ~ is odd. Which elements of A may occur as V(L)/(t- I) ~-]
for a ~-component
link L all of whose linking numbers are 0? If all the linking numbers of a ~-component 9 3, then Kidwell's
theorem implies that AI(L )
and so AI(L) is in I ~+] .
link are 0, and if
-_
(t i - I)
is in I,
Can this exponent be improved upon?
has announced that AI(L) is in 12~-3.
Traldi
(~192~, See Chapter VII).
have not been able to find a 4-component
We
link, all of whose linking
numbers are 0, for which the first Alexander polynomial is not in the 6th power of the augmentation
ideal.
108
Null Reduced Ideals
Definition
The reduced nullity of L is K(L)
=
min{k[#(Ek(L)) # 0}.
It is obvious that I .< ~(L) .< K(L) .< ~
and it is easily checked
that K(L) = 1 + rank A HI(XT; 7/).
Theorem 3
The following are equivalent:
(i)
K(L) = ~ ;
(ii)
the longitudes of L are in (ker~)', the c o m u t a t o r
(iii) HI(X;Z~) ~
HI(W; 2Z) and HI(X~ ; 2Z) ~
subgroup of k e r ~ ;
HI(WT; ZZ), where W is as in
Theorem 2 above. Proof
If L is a v-component sublink of L, then K(L) - K(L) .< ~ - V ;
K(L) = v
if K(L) = ~.
hence
Since the Torres conditions imply immediately
that a 2-component link with reduced nullity 2 has linking number 0, so all the longitudes of L are in G' C ker T. 9
l
th
longitude in HI(XT ; 2Z) ~
Let %i be the image of the
ker T/(ker ~)'.
Then (t- I).~. = 0 since 1
each longitude conmmtes with a meridian.
By an easy argument as in
Theorem VI.| there is a d E ~(E (L)) such that e(6) = 1 and which annihilates the A-torsion submodule of HI(XT; Z~);
hence Ann(hi) ~_ (t-l,~) =A.
Therefore all the longitudes are in (ker T)'. Conversely, all the longitudes are in G' if and only if H I (W; 2Z) = ZZ~ , in which case there is an exact sequence 0 --+ H2(xT; 2Z) --+ H2(WT; 7Z) --+ H2(WT,XT; 2Z) --+ ~ HI(XT; ZZ) --+ HI(WT; 2Z) --+ 0. By excision, the middle A-module is isomorphic to 7z~ , and the boundarymap a is trivial if and only if either (ii) or (iii) is true.
Thus (ii) and
109
(iii) are equivalent.
The remaining implication now follows on taking
rational coefficients Q and appealing to duality, which implies that H2(WT;Q) ~ Q (~ (QA) r where r is the rank over QA of H I ( W T ; Q ) ~ HI(XT;Q) and so equal to K ( L ) - i . // By an argument using the Universal Coefficient spectral sequence and Poincar~ duality, again as in Theorem VI.I, it may be shown that (if K(L) = ~) the ideal ~(E (L)) is principal if and only if the A-module HI(XT; 7Z)~ HI(WT; 2Z) has a free summand of rank ~-I. the case for an homology boundary link. onto A ~, then ~(E (L)) = (~A (L)).
This is always
More precisely, if A(L) maps
For E~(L) = (A (L))Eo(B) by
Theorem VI.I, and ~(E (L)) is a principal ideal (by the above remark) which is not contained in (t-I). ~(Eo(B)). (t- I)N
But
-~(t i - I) I,
for N large.
Therefore the same is true of annihilates B, so ~(Eo(B)) contains
Therefore ~(Eo(B)) = (I).
However for the
2-component link L = L I U L 2 of Figure V.I, ~(E2(L)) = (3,1+ t) 2, and so this link is not an homology boundary link.
Fibred Links Definition
A ~-component n-link L is fibred if there is a fibre bundle
projection ~ : X § S I such that the induced map of fundamental groups is the total linking number homomorphism. If ~ = I, the final clause is redundant, and each fibre of ~ is a Seifert surface for the knot.
If n = I, the example of the link
which has exterior homeomorphic to S 1 x S 1 x E0,|]
O O
shows that the exterior
may fibre over the circle in many ways, although there is essentially only one fibration satisfying the condition on meridians.
We shall show
110
below that when n is greater than 1 there are no fibred n-links with more than one component. Let e : IR § S I be the exponential map, sending r in ~
to
e(r) = e 2~ir in S1 , regarded as the unit circle in the complex plane. Since ]R is contractible,
the pullback e*~ is a trivial bundle,
and so
there is a commutative diagram E ---+ X
F x ]R
e
IR
where F = ~-i(I) factor,
~
is the fibre of ~, pr 2 is projection onto the second
and E is a covering map.
determines generates
a homeomorphism
The translation
and determines
only defined up to isotopy, and corresponds
The map h is called the characteristic it up to isomorphism.
Although h is
the induced map on homology
is well defined,
to multiplication
by t on the homology of the infinite
cyclic cover F •
of X, considered
compact orientable
(n+l)-manifold
connected
since the map
the construction Theorem 4
The fibre F is a
with ~ boundary n-spheres,
: G § ~
is onto.
and is
(For more details on
homeomorphism
see ~133; page 67].)
Let L be a fibred ~-component n-link
(ii) If n > 1 then
~ = I.
~ 0. In other words, L is an n-knot.
As an abelian group HI(XT ; ~)
isomorphic if n = I ,
~i(~)
as a A-module.
of the characteristic
If n = 1 then Ared(L)
PrOof
r ~-+ r + 1 of ]R
h of F such that H : (v,r) +-+ (h(v),r+ I)
the covering group of Eo
map of the bundle,
(i)
S1
to HI(F ; ~).
Ared(L ) # O.
Therefore
is finitely generated,
since it is
it must be a torsion A-module and so,
111
If n > I then Stallings' Theorem and Theorem Vol imply that E _I(G) = 0 and so HI(X' ; ~) has rank ~ - I as a A~-module.
Since the
boundary maps in the equivariant chain complex for X T may be obtained from those for X' by the change of coefficients ~ : A HI(XT; ~) as a A-module is at least B - I.
§ A, the rank of
Hence it can only be finitely
generated as an abelian group if ~ = I. // In view of this theorem we shall continue to concentrate on the case n = [.
(For a survey of results on higher dimensional fibred knots,
see Section 5 of [98]).
The linking number of a fibred 2-component
link is unrestricted, as the Whitehead link (~ = O) and the (2, 2~)-torus link (s + O) are fibred.
A fibred 2-component link has
linking number 0 if and only if the boundary of the fibre is a union of longitudes, for the linking number is the image of either longitude under the total linking number homomorphism when ~ = 2.
Goldsmith has
observed that the (3-component) Borromean rings are fibred E55]. Since the action of h determines the Al-module structure on HI(X;AI) = HI(F ; ~), the characteristic polynomial of h I = Hi(h; ~) generates Eo(HI(X;AI)) while the minimal polynomial of h I generates Ann HI(X;AI).
Therefore if B = ] the characteristic polynomial is AI(L)
and the minimal polynomial is %I(L) = AI(L)/A2(L), while if ~ > ] the characteristic polynomial is (t - ])Ared(L) and the minimal polynomial is (t-I)Ared(L)/AI(HI(X;AI))
which divides (t-I)~(AI(L)/A2(L))
since
EI(HI(X;AI)) = ~(E2(L)) is contained in (~(A2(L))). The most interesting class of (fibred) links are those associated with an isolated singularity of a plane algebraic curve~
Let f(w,z) be
a polynomial with complex coefficients which vanishes at the origin and suppose that f has at most an isolated critical point there.
If E > 0 is
i
so small that the ball B e = {(w,z) I lwl2 + [zl 2 ~ e } contains no critical
112
point of f other than the origin, then the pair (S~ ,S~
n f-l(0))
determines a fibred link whose link type is independent of go
(The
fibration is given by ~(w,z) = f(w,z)/If(w,z) I for (w,z) in S e3 _ f-1 (0) E133; page 53 .)
We shall call such a link an algebraic link.
The
geometry is well understood, as the link is an"iterated torus link" and is specified completely by the Puiseaux expansions of the U irreducible branches of f at the origin each irreducible branch.
E],112,1843o
(There is one component for
Note that Milnor and others use ~ to denote a
different invariant of the singularity E133; page 593o)
The action of
the generator of ~I(S I) on the homology of the fibre HI(F ; ~) via h I is called the (local algebraic) monodromy of f at the origin.
In the
remainder of this section we shall list some of the more striking results on the monodromy.
(We remark that most of these results extend to the
higher dimensional case of isolated singularities of hypersurfaces in cn+l, although the proofs in some of the references quoted below depend on the geometry of the classical case.) Since an algebraic link is an iterated torus link, the roots of its reduced Alexander polynomial are roots of unity, and so the monodromy is quasiunipotent, integers m and N
in other words, (hm! _ ])N = 0 E15,112,1843.
for sufficiently large
L~ showed that when ~ = ] the monodromy
is of finite order, equivalently that the minimal polynomial II(L) has distinct roots EI|2], but this is not true in general.
A'Campo has
shown that the link of (w2 + z3)(w 3 + z 2) has monodromy of infinite order Eli . Durfee showed that the characteristic polynomial is exactly divisible by (t-I) p-I, in other words that ?(n)(1) # 0 E46]o
In
particular a 2-component algebraic link has nonzero linking number, while the Borromean rings cannot be algebraic.
He showed also that the
113
monodromy is of infinite order if (t + I) ~ divides the characteristic polynomial.
This criterion was extended by Sumners and Woods who
showed that if ~ is a pm th root of 1 for some prime p and positive integer m, and if ( t - g ) ~ divides the characteristic polynomial, then the monodromy is again of infinite order E1843. A'Cemposhowed that the trace of the monodromy is ], and deduced that the connected sum of algebraic knots is never algebraic
E23.
L~
showed that concordant algebraic knots are isotopic EI123. Brieskorn indicated how the complex monodromy h I O
r could (in
principle) be derived algebraically from the function f E153
Although
the complex monodromy does not determine hl, it does determine the characteristic polynomial of h I .
In another approach which seems more
workable, A'Camp~related the characteristic polynomial to a Zetafunction E3~.
Durfee has shown that there is a basis for HI(F; ~) for
which the Seifert matrix is upper triangular with entries -I on the diagonal
[45].
Finite Cyclic Covers Below the total linking number cover
q : X T § X lie the k-fold
cyclic covers ~k : Xk § X corresponding to the subgroups T-l(k ~) of G. The space X k is just X T modulo the action of tk, and there is an infinite cyclic cover X T + Xk.
Thus the homology of the finite cyclic covering
spaces may be determined (up to an extension problem) from that of the total linking number cover via Wang sequences.
Conversely Cowsik and
Swarup have shown that the integral homology of X T imbeds in the inverse limit of the homologies of the X k [35]. be true of the rational homology.
(They note that this need not
For example, consider any knot, such
114
as 41, for which the Alexander polynomial is nontrivial and has a noncyclotomic factor~ of Gordon E58]
They have thereby simplified and extended arguments
and Durfee and Kauffman
E47]
on periodicity in the
homology of branched covering spaces of knots. The finite cyclic covers of the exterior of a link may be completed to branched covers of the sphere.
(See E|57; page 292]).
this is a fruitful way of constructing
When n = !
closed 3-manifolds.
Branched
cyclic covers are also central to the most natural proof of the invariance of the Hermitean signatures of a link under concordance E88,
198]. The Wang sequence for the cover X T § X k in low dimensions reduces
to --+ HI(XT; ~)
tk-I 3 , ~ HI(XZ; ~) --+ HI(Xk; ~) --+ ~
--+ 0 .
The image of the k th power of a meridian in HI(Xk; ~) generates a submodule mapped isomorphically
onto ~
by 3, and so HI(Xk; ~) ~
where Ck = HI(XT ; ~)/(t k - I ) H I ( X T ; ~)o
~
Let M k = X k 0 ~D 2 • S 1
k-fold cyclic branched covering space of a l-link Lo
~
be the
Then Shinohara
and Sumners show that the map H2(Mk,Xk; ~) + HI(Xk; ~) is injective Usingthis Sumners computes the Betti numbers of Mk D 8 3 ] o knot (~ffi I), then HI(Mk; ~) ~
Ck,
[172].
If L is a
Ck, and Weber shows then that C k is finite,
of order IRes(t k - I,AI(L)) I = [ ] - - ~ 0$i
Al(L)(~i)l where ~ is a primitive
k th root of unity, if and only if this number is nonzero ~ 0 3 ~ .
Since
IAI(L)(])I = I for a knot, this number may be rewritten as IRes((t k - |/t-I),AI(L)) I = I { I l$i
AI(L)(~i)I 9
for ~l(Xk) derived from a Wirting~presentation
Using a presentation
for G by a Reidemeister-
115
Schreier process, Kinoshita shows that in the latter formulation the assertion remains true if ~ > I.
ElO4a].
that for the 2-component (2,4)-torus link
(Note that he has observed ~
the first
homology o f M 2 i s ZZ/4ZZ, and is not a direct sun,hand of that of X2, which is ZZ2~ZZ/2ZZ.
A late addition.
See [172]).
Sakuma has given a neat calculation of the order of
the homology of a cyclic cover of S S, branched over a link [209].
By
the Cartan-Leray spectral sequence for r:X' -> X r, the sequence A 0 --+ R(=r,HI(X')) --+ HI(X r) -~ 77~-I (= TOrl ~ (AI, 77))--+ 0 is exact.
Sakuma observes H I(Mk) = H I(XT)/vH I(XT), where v = (tk - I/t - I),
and hence that there is an exact sequence 0 --+ N/~R --+ N 1 (Mk) --+ (ZZ/kFZ)~-| --+ O. If V(L) + O, then H 1 (XT) has no pseudozero submodule and (hence) R has a square presentation matrix.
By Lemma 111.6 and page 102, &o(R) = V(L).
By the argument of Weber [203J, the order of R/~R is ]Res(V(L),~) I and so that of HI(M k) is IRes(Areal,v)I .
(Note that [Nes(t-1,~)l = k).
Hl(Mk) is finite if and only if this number is nonzero. of [c(V)[ is equally neat.
Clearly
Sakuma's calculation
The Wang sequence for q :XT --+ X leads to
H2(X) --+ HI(X ~) -+ R --+ 0 (as R = (t-l)H l (XT), by an earlier sequence) and hence to H2(X ) _~U Hl(XX)/(t - I)HI(X T) (= ker
B:Nl(X) --+ N (XT)) --+ R/(t-I)-~ o
where the matrix of U with respect to suitable bases is a minor of the matrix N of page I05, so le(v) l = ]Ao(R/(t-l)R) [ = ]El(N) I.
0
CHAPTER IX
LOCALIZING THE BLANCHFIELD PAIRING
A (2q-l)-knot is determined up to concordance by the equivalence class of its Blanchfield pairing in a certain Witt group, if q > 2 ~4,97,118,180].
(Kervaire had shown earlier that all even
dimensional knots are null concordant ~96J).
This pairing is also
defined and of considerable interest in the classical case q = I, although it is no longer a complete invariant of concordance
[24].
In this chapter we shall propose similar invariants for classical links. The Blanchfield pairing on the quotient of the A -torsion of the Alexander module of a l-link L by its maximal pseudozero submodule, after localization with respect to S, the multiplicative system in A generated by the elements tl-l,...t -I , is a primitive Hermitean pairing into Q(tl,...t )/A S.
We shall show that the class of this pairing in
the Witt group of such pairings depends only on the concordance class of L.
If local knotting is factored out, by passing to the coarser
equivalence relation of weak concordance, generated by concordance and isotopy, the
appropriate invariant is obtained by localizing further
with respect to the multiplicative system generated by all the nonzero polynomials in one variable.
(This more thorough going localization
was motivated by Rolfsen's result that the "Alexander invariant" localized at E is invariant under isotopy [155]). These invariants may be computed for a boundary link from a Seifert matrix for the link.
The invariant of concordance specializes to the knot
concordance invariant of E9~] for l-component links; the weak concordance invariant is clearly trivial for knots.
For 2-component links, the
coefficient ring A E is a principal ideal domain, so a primitive AE-torsion pairing is perfect, and the Witt group of linking pairings over A E can be expressed as a direct sum of Witt groups of Hermitean forms over involuted
117
fields.
We shall show that the image of the set of 2-component
boundary
links in this Witt group is a subgroup which is not finitely generated. In the next section we refer briefly to several definitions
of signatures
for links, in order to raise the question as to how they are related to the Witt classes introduced here.
We conclude
in which we use a surgery description
this chapter with an appendix
of a knot in order to describe its
Blanchfield pairing.
Linking Pairings on Torsion Modules Let R be a corm~utative noetherian and with field of fractions F = R . o denote the involution
domain with an involution -:R + R, We shall use the overbar also to
induced on F and on the R-module F/R.
Let ~ = • I.
(There will be no risk of confusion with the augmentation homomorphism
in
this chapter). Definition
A map c:N 1 x N 2 + N 3 of R-modules
if it is ~ - b i l i n e a r , the second variable
R-linear
in the first variable and R-antilinear
in
(that is, c(rnl,n 2) = rc(nln 2) = c(nl,~n 2) for all
n I in NI, n 2 in N 2 and r in R). (respectively,
is s ses~uilinear pairing
right)
The pairing c is primitive
on the left
if for each nonzero n I in N 1 (respectively,
there is some n 2 in N 2 (respectively,
n 2 in N 2)
n I in N I) such that c(nl,n 2) + O.
An s-Hermitean pairing on a torsion module M is a sequilinear pairing b:M • M + F/R such that b(m,n) = ~ b(n,m) adjoint of b is the R-antilinear Ad(b)(n)(m) if Ad(b)
= b(m,n)
is injective
for all m and n in M.
map Ad(b):M + HomR(M,F/R)
for all m and n in M. and perfect
if Ad(b)
If N is a finitely generated R-module submodule of N we shall let ~ N = tN/P. tN = ~ N for any R-module N).
The
defined by
The pairing b is primitive is bijeetive.
and P is the maximal pseudozero
(Of course if R is a PoI.D.
As remarked in Chapter
then
III, if a torsion
118
module M supports a primitive bilinear
(or sequilinear) pairing,
have no nontrivial pseudozero submodule, Lemma ]
it can
so M ='~M.
If R is a principal ideal domain, then a primitive e -Hermitean
pairing on a finitely generated torsion module M is perfect. proof
By the structure theorem for finitely generated modules over
P.I.D,s, M is a direct sum of cyclic modules, (noncanonieally)
isomorphic to M.
and so HomR(M,F/R)
is
As M has finite length, any injective
(antilinear) endomorphism is easily seen to be bijective. //
Definition
An e-linking pairing over R is a finitely generated torsion
module M with a primitive e-Hermitean pairing b. pairings
(M,b) and(M',b')
module M ~ M '
is the pairing
The sum of two such
(M,b) ~D (M',b') with underlying
and with map sending ( m ~ ) m l ' , m2 ~ m 2 ' )
b(ml,m 2) + b'(ml',m2'). N such that N = N ~
to
A pairing is neutral if M contains a submodule
= {m in Mlb(n,m) = 0 for all n in N}.
(M,b) and(M',b') are Witt equivalent
Two pairings
if there are neutral pairings
(N,c)
and (N',c') such that (M,b)@ (N,C) = (M',b') ~D (N',c').
Proposition-Definition
The set of Witt-equivalence
classes of e-linking
pairings, with addition defined by sum of representative
pairings, and with
(M,-b) representing the inverse of the class of (M,b), is an abelian group, denoted We(F,R,-).
//
The Witt groups of greatest interest to the algebraist are based on perfect pairings with an additional quadratic structure.
If 2 is invertible
in R, any perfect pairing can be endowed uniquely with such a quadratic structure,
and so the relative Witt group Woe(F/R) of Pardon
embeds in Wc(F,R,-)
~148] then
(since in any case the definition of neutral is the
119
same for primitive and perfect pairings).
All these distinctions
vanish
when R is a P.I.D. containing 89 as in the ring A21 discussed below in connection with 2-component
links.
For the remainder of this section we shall assume that R is factorial. lena
2 (Blanchfield)
If c:N 1 • N 2 + F/R is a sesquilinear pairing
which is primitive on both sides, then (Ao(N1)) = (Ao(N2)). proof
This is an immediate consequence of Theorem 4.5 of Blanchfield
Alternatively, system R - ~ ) ,
Eli].
it may be proven by localizing with respect to the multiplicative for each height
| prime~
of R.
Compare our proof of
Theorem VII.]. // lemma 3
If (N,c) is a neutral e-linking pairing over R, then (Ao(N)) = (ff)
for some nonzero f in R. proof
By assumption there is a submodule P C N such that P = P!.
The
pairing c induces a sesquilinear pairing of P and N/P into F/R which is primitive on both sides.
By the preceding l e m a
(Ao(P)) = ( A o ~ ) ) .
Therefore by lemma 111.6 (Ao(N)) = (Ao(P))(Ao(N/P))=(Ao(P))(Ao(P)).
Let
f be any generator of (Ao(P)).// Theorem 4
If (M,b) and (M',b') are Witt-equivalent
e-linking pairings over
R then there are nonzero elements f and f' in R such that (ffA (M)) = (f'f'Ao(M')). o
Therefore the map sending (M,b) to the class of A (M) modulo O
{ugglu in R , g in F } induces a homomorphism from We(F,R,-)
to ~ ~
F /R ) =
{vf[v in R , f = ~ in F }/{ugglu in R , g in F }. proof
By assumption there are neutral pairings
(M,b) ~
(N,c) ~ (M',b')(~) (N',c').
lemmas.
The rest is clear. //
Definition
(N,c) and (N',c') such that
The first assertion now follows from the
The Alexander class of (the Witt class of)
(M,b) is the image 6(M) of ~ (M) in ~~ O
;F* /R * ).
an e-llnking pairing
120
We shall let K of A . and by
(or K) denote the field of fractions Q(tl,...t ~)
The multiplicativesystems
in A
generated by {tl-l,...t -I}
U ~ [ti,t.-lJ-{O} are denoted by S and E respectively. l
Blanchfield Duality In this section we shall give an explicit geometric description of a version of equivarent duality, and use it to define our invariant. Blanchfield showed that if p:~ + X is a connected cover of a compact oriented n-manifold X, with covering group ~ ~, then there is a sesquilinear pairing V:~H (~) • ~H (~, ~ ) p n-l-p sides Ell].
+ K~A which is primitive on both
(Here and below in this chapter we shall let H,(Y) denote the
integral homology of the space Y).
If x is a p-cycle of X such that
~x = ~u for some (p+l)-chain u and some nonzero ~ in A, and y is a relative (n-l-p)-cycle
such that BY = 3v for some (n-p)-chain v and some nonzero
8 in A, then V(x,y) = ~-Is(u,y) = ~-Is(x,v) where S(c,d) = and I is ordinary intersection of chains.
E in ~
I(c,Xd).X
From this definition,
it is not
hard to see that if n = 2p+l, then V induces a (-l)P+1-Hermitean [ , ] on ~H (~) by Ix,y] = V(x,j,y) P whic~ however may not be primitive.
(where j is the inclusion
and
A.
Now let L be a ~-component
l-link with complement X, and let p:X' + X
be the maximal abelian cover of X.
Then the meridians of L determine an
isomorphism of the covering group with ~ ~. pairing is a map ~HI(X' ) • ~HI(X',~X' ) ~ K/A.
Then the Blanchfield
In general however,
linking
In the c~se of knots HI(X')
is a Al-torsion module and the natural map HI(X') § HI(X~3X') isomorphism.
(X,~) + (X,~X))
He showed also that intersection numbers
give a nonsingular pairing of the "Betti" modules Hp(X)/tHp(~) Hn_ p(X, ~ ) / tHn_ p(~, ~ ) into
pairing
is an
the two outer maps of the following part
121
H~(~x') §
H~(X') § H~(x',~x') § Ho(~X')
of the long exact sequence of the pair (X',~X') are non-zero. Nevertheless HI($X') and H0(~X') vanish after localizing with respect to S (and, afortiori, with respect to l), since both are quotients of
Thus the localized Blanchfield pairings
~-'-]s : tHs(X )S • tNi(X )S
K/AS
and
[-'-]E : tH~(X
• till(X')Z § K/Az
are primitive, (+l)-Hermitian pairings.
(Notice that the As-tOrsion of
M S is the localization of the A-torsion of M, so the notation tM S is unambiguous. Definition
Note also that A(L) S = H I ( X ' ) s ~ A S and ~HI(X') S = IA(L)s). For L a ~-component link, Bs(L) (respectively. BE(L)) is
the class of ~HI(X')s,
t [-,-IS ) (respectively, (tHI(X)Z'
[-'-]I )) in
W+I(K ,A S,-) (respectly, W+I(K ,A Z,-)). For knots multiplication by | - t induces an automorphism of HI(X') [96] and so essentially no information is lost on localization with respect to S in that case.
Localization with respect to Z annihilates HI(X')
for X' the infinite cyclic cover of a knot complement; that is lost;
but that is all
Rolfsen showed that any PL isotopy of a link could be
achieved by introducing or suppressing local knots, and that hence localizing with respect to Z the homology of the maximal abelian cover of a link complement gave an
isotopy invariant [155].
It is easy to see that
the Z-localised Blanchfield pairing is also invariant under such local isotopy, and hence under isotopy.
122
Computation from a Seifert Matrix The computation of the Blanchfield pairing from a knot in terms of a Seifert matrix, as done byKearton ~93], may also be carried through for boundary links.
Let L : ~S I § S 3 be a boundary link, and let Uj,
I < j < p, be p disjoint orientable surfaces spanning L.
Then
orientations for the surfaces U. compatible with those of the L. are J J determined uniquely by the convention inward normal last (equivalently, by insisting that for each j the orientation class FUj,~Uj] in H2(U j,~Uj) map to [Lj] in H I (~Uj) in the long exact sequence of the pair (U.,~U.)). Let Y = X - ~ W. where the W. are disjoint open regular J ] i=I J J neighbourhoods of U. in the link complement X, ] < j < B. There are two 3 natural embeddings of each U. in Y; call the one for which U~ is compatibly J J oriented with ~Y i. and the other i . . Then the module HI(X') is J+ ..1contained in the following segment of a Mayer-Vietoris sequence: dI HI(U) ~)7zA ----+ HI(Y) ~ ) ~ z A §
dl § Ho(U)~)Tz A---+ Ho(Y) ~)TzA -+ 7z -',- 0
where U =
U U. and d, I H , ( U j ) ( ~ A j=l 3
Lemma 5
~HI(X') = tHI(X) = Coker d I.
Proof
= (ij+),Otj
- (ij),~
Clearly tHI(X') .~c Coker dl, since Ho(U) ~ ) A is free.
I.
On localizing
the above sequence with respect to K, we obtain the exact sequence
O -~(Coker d I) ~ A K ~ HI(X') O A K * K p + K + 0. On the other hand a similar localization of the Crowell sequence relating HI(X') to the Alexander module g~ves
0 ~ H I ( X ' ) O A K + A(L) ~ A K + K + 0 and A ( L ) ~ A K = K p for a p-component boundary link (since then E _I(L) = 0 and E (L) # 0). P module.
Hence (Coker d I) ~)A K = 0 and so Coker d I is a torsion
123
Let {~jm [ 1 ~ m ~
m(j)} be a basis for HI(Uj) , 1 < j < ~.
By
Alexander duality, the ordinary linking pairing s in S 3 [177; page 361~ establishes a duality between HI(U) and HI(Y) = HI(S3-U);
let
{~jm [ 1 ~ m 4 m(j), l ~ j 4 ~}be the dual basis, so that Z(ejm' ~kn ) = 6jk~mn"
Let A be the matrix of
(i+), = j__~l(ij+), : El(U) + HI(Y) with respect to these bases; then d I is represented by the matrix A = @A - A tr where @ is the diagonal matrix
diag (tllm(1), ... t ~m(~)) ,
and det g ~ O since it generates Eo(Coker dl).
Then there is an exact
sequence O +AM+
A M ~ t H I ( X ') + O
(where M =
E m(j)). Therefore by Theorem III. I0 tHI(X') has no j=1 nontrivial pseudozero submodule, and so t'~HI(X') = t(l~l(X'). // Using the same symbols to denote l-chains in U, Y representing the classes ~jm' ~kn respectively, consider the 2-chain [-I, I] x ~jm in X'.
Then ~([_-1,1~ x ~jm)=tjij+(~jm) - i.j_(~jm) and so ~I~-1,1~ x E qjm~jm] A
^
represents E (Aq)kn=kn.
Therefore, if ~ = det A, ~ 9 ~ rkn~kn is the
class of ~ [L- 1,1] x E (A-16" r) jmajm I , where the matrix A -i "6 has coefficients in A, and so the Blanchfield pairing on tHI(X') is given by ~(r), *(s)] = V(*(r), j,0(s)) = ~- S
i,i] • Z (4
~r)jm~jm , ~ Skn~kn
^ = ~l I -Sen Z (A-~6r)jmS( E-l,l] x ~jm,~kn) l
=
-
~ E Skn Z (A-16r)jm(|-tj)6jk6mn = str(l-O).A-ir mod A
(where r and s are columm vectors in AM).
Thus for boundary links the
Hermitian pairing on ~HI(X') is primitive, and in fact perfect, even before localization.
(Notice that detA
represents the Alexander class here).
124
A similar construction works
for homology boundary links,
for
in [73] it is shown that splitting along singular Seifert surfaces V =
U V. for such a link leads to a short free resolution 1~i~ ~ (HI(V)/HI(~V))
Q A § (HI(X-V)/HI(~X-V))
where B is the submodule By Theorem
~ A§
§ 0
generated by the image of the longitudes.
VI.2, the module
tHI(X')/B is ~HI(X').
better to localize and thus have an invariant for the set of (homology) boundary
It is however
applicable
to all links,
links is not closed under concordance,
as is shown by the examples of figures V.| and VI.I.
Invariance Under Concordance The above construction
of the Blanchfield pairing for boundary
links in terms of Seifert surfaces
can be used to show that
Bs(L) = BS(~) if there is a concordance ~ from L to ~ with group =~I(S 3 x I - im ~) such that 5 / > m is freely generated by a set of meridians,
by imitating Levine's proof that when ~ = |, Bs(k) depends
only on the concordance
class of the knot k [I18].
in general that every concordance between boundary "boundary concordance" concordant
to a boundary
in this sens~
links is a
let alone that every link
link is a boundary link, an argument which
does not rely on Seifert surfaces Theorem 6
As it is not true
is to be preferred.
Let L 0 an d L I be concordant
~-component
links.
Th~n
Bs(L O) = Bs(LI). Proof to L = ~
Let ~ I ~SIx
the image of -r
: ~S I x I § S 3 x I be a concordance {l}.
from L O = d I~S I x {O}
Let N(~) be an open regular neighbourhood
S 3 x 1, and let Z = S 3 x 1 - N(~).
Then
for
125
~Z = X 0 U H(slxslx]) U X 1 ~ X 0
U
~SlxSl
X 1 where X. = Z n S 3 x {i} is l
complement of L. (for i=O,]) and where the jth boundary component of i X 0 is identified with the jth boundary component of X 1 via an orientation reversing map.
The inclusion X 0 + Z, X 1 § Z
isomorphisms on homology.
each induce
On localizing the Mayer-Vietoris sequence
' 1 ') with respect to S, it follows that of the triple (~Z',Xo,X HI($Z)s = H I ( X 0 ) s ~ H I ( X I ) S.
Clearly the Blanchfield pairing on ~HI(~Z) S
is the direct sum of the Blanchfield pairing on tHI(Xo) S with the negative of the Blanchfield pairing on ~HI(XI) S.
Thus to show that Bs(L O) = Bs(L I)
it will suffice to show that IHI(~Z') S contains a submodule equal to its own annihilator with respect to the pairing. !
A. Kawauchi showed in ~90; lemma 2.]] that H2(Z',X 0) is a torsion A-module, and hence that the image of H2(Z') in H2(Z',~Z') is contained in the torsion submodule.
Therefore the sequence
tH2(Z' , ~Z') ~ tHI(~Z') § tHI(Z') is exact.
Let P = ~[tH2(Z',~Z') ] .
Then the image of PS in ~HI(~Z') S is such a submodule. For let Q, R be relative 2-cycles on (Z',~Z') representing torsion classes in H2(Z',~Z') and let q, r be the boundaries of Q, R respectively,
]-cycles on 3Z' representing classes in P.
for some nonzero e in A and for some 2-chain s on ~Z'. ]
Then ~r = ~s Then
~
^
V~z,(q,r) = ~1 Z 13z,(q,ys)y = =~ Z Iz,(Q, y~) = Vz,(Q,r) (where r, Y denote r, s considered as chains on Z') = 0 in K/A since ~ bounds R in X'.
Thus P C Pm.
Now let w be a ]-cycle on 3Z' representing
a torsion class in tHI(~Z')
(so that 8w = ~W for some nonzero 8 in A
and some 2-chain W on 8Z') and suppose V z,(q,w) = 0 for every ]-cycle q as above, representing a class in P.
Then Vz,(Q,~) = V~z,(q,w), and
hence by the primitivity of the Blanchfield pairing for (Z',~Z'),
126
bounds in Z'. d, and so ~
Hence the class represented by w is in the image of = P.
theorem is proven. Corollary
It follows immediately that P ~
= PS' and so the
//
If L is a null concordant link, Bs(L) = O.//
The analogous results for BE(L) are also immediate consequences
of
this theorem, namely B~(Lo) = BE(LI) if L O is weakly concordant to LI, and in particular BE(L) = 0 if L is weakly concordant to the trivial - component link. Corollary
Let ~s(L) denote the least principal ideal in the U.F.D.
A S containing Eo(t(A(L)) S = Eo(tHI(X(L)')S). if L O and L I are concordant,
Then ~s(L) = ~s(L) and
then there are nonzero fo' fl in A S such
that fo.fo-6s(L O) = fl.fl-6s(Ll).
(Similarly for localization with
respect to E). Proof
These assertions are consequences of the discussion above of the
determinant of a linking pairing,
and of the result Bs(L O) = Bs(LI). //
The ideal 6s(L) is generated by the image of the first nonzero Alexander polynomial of the link.
Kawauchi has proven the stronger
result that if L O and L 1 are concordant
links, then there are nonzero
go and gl in A with ~(go ) = e(gl) = I and such that go-go.A (Lo) = gl.g1"A
(e I) (where ~ = ~(eo) = ~(el) ) D0]
Additivlty Let L_ , L+ be D-component
links.
After an ambient isotopy of each
link, it may be supposed that imL_ C D 3,_ imL+ C D 3+ (where S 3 = D 3_ U 2 D~) S and that for each i, ] ~ i ~ D, the i th component of imL
meets imL +
only in an arc contained in the i th component of imL+, which receives opposing orientations
from L
--
and L . +
Then the closure of
127
imL - - u imL +
-imL - - N imL +
is the image of a compatibly oriented
link.
If ~ = 1, the ambient
isotopy type of this link is well defined by the ambient of L_ and L+. ~-component C 1 = C~
Let C~ denote the set of concordance
links.
Then this connected
the set of concordance
homomorphism
from C I to W+(Q(t),
true in general
abelian link) connected concordant ways. Figure
so that C 1 becomes
pairing gives rise to a
~ It,t-l],- ) [94, llS].
that the connected
even modulo concordance,
classes of
sum induces an addition on
classes of knots,
an abelian group, and the Blanchfield
isotopy types
It is not
sum of two links is well defined,
Already with L_ = -L+ = C O
(the two component
sums may be formed in at least two non-
The link of Figure
lb has first Alexander
are not even I-equivalent
]a is trivial, whereas
ideal nonzero,
and so these two links
.
(a)
(b) Figure
l
the link of
128
Furthermore,
the Alexander module of the link of Figure 2, which is
a connected sum of the abelian link with itself,
is A/(]+tlt2)
which when
Figure 2
localised with respect to I has length
] as a A -module,
and so can
not support any pairing whose Witt class is divisible by 2, as would be expected if the Blanchfield modules admitting Nevertheless
pairing were always
'additive',
for
such pairings necessarily have even length.
some additivity
results may be obtained by restricting
the classes of links considered.
Let L I ~ L
2 denote any link formed
in the above fashion from two links LI, L 2 : ~S 1 ~ S 3. In the first place,
if ~ : ~S 1 x I * S 3 • I is a concordance
-~ I P* x I embeds p disjoint the isotopy extension im~
is contained
theorem
particular
~59;
page 56~ it may be assumed that
to L~ and L~ respectively,
is concordant
classes containing
Hence if
then any link of the
to some link of the form L ~ = L ~ .
the set of concordance
then
and so by general position and by
in D 3 x I and meets S 2 • I in (p arcs) • I.
L 1 and L 2 are concordant form L I ~ h L 2
arcs
In
split links forms
129
an abelian group, isomorphic to (CI)~, which acts on C~.
In fact C~
is (Cl)~-equivariantly isomorphic to (CI)~XC~, where ~I is the set of weak concordance classes of ~-component links.
(The map ~
+ C~ sends
the weak concordance class of L to the concordance class of L#/~[ ~ (-Li)]
2 is a boundary link with a
system of Seifert surfaces given by U l i ~ U 2 i .
It is then clear from
the interpretation of the Blanchfleld pairing in terms of Seifert surfaces that Bs(LI~#~L 2) = Bs(L I) + Bs(L2).
(Not every connected
sum of boundary links is a boundary link though, for the ribbon link of Figure 3, which is not even an homology boundary link, is the connected sum of two copies of the trivial link.
Figure 3.
130
For this link, and for the trivial link, the Alexander module is torsion free, and so the Blanchfield pairings still add.)
This shows that
the image of the class of boundary links in W+l(K, AS, -) is a subgroup (Since Bs(-L) = -Bs(L) it is clearly closed under taking inverses). Thirdly, if L 1 and L 2 are ~-component links with Alexander nullity ~(L I) = ~(L 2) = ~, then ~ ( L I ~ = L 2 )
= ~ also.
(More precisely, if a
link ~ is obtained from a link L by one saddle point amalgamation
~49]
then an examination of Jacobian matrices shows that ~(~) ~ e(L) - I. Hence ~ ( L I ~ L 2 ) links).
~ ~(LI) + ~(L2) - ~ if L 1 and L 2 are both ~-component
This is of some interest as the set of all such links contains
all (homology) boundary links, and is closed under concordance by Theorem V.2.
Is every link L with ~(L) = ~ concordant to a boundary link?
If L I is a boundary link in D --3 (as above) and L 2 is a link in D +3 with ~(L2) = ~ which meets L I in ~ disjoint arcs in S 2 = D 3_ N D~, then it can be shown that B s ( L I ~ L 2) = Bs(L I) + Bs(L2).
Is it generally true that
the Blanchfield pairing is additive for links L with ~(L) = ~?
2-component Links In this section we shall specialize to the case ~ = 2, and consider there the E-localized pairing.
As the coefficient ring is then a P.I.D.
the algebra simplifies greatly.
Firstly tHI(X )E
tHI(X') ~ and the
Blanchfield pairing is perfect, for the Universal Coefficient spectral sequence and Poincare duality then give an isomorphism.
HOmA (tHI(X')E, K/A E) ~ E x t ~
HI(X')E,A E) ~ tHl(X') E
which is just the inverse of the adjoint map of the Blanchfield pairing. (In fact a more careful examination of the spectral sequence over QA S shows that the Blanchfield pairing on Q ~ ) t H I ( X ' ) S is already perfect).
131
Secondly if M is a finitely generated torsion module over a P.I.D. R then M = ~ ~
(summation over nonzero primes ~ of R) where
%
PM~ =
~) (R/~ei) is the ~-coprimary submodule, and ~ + M l~i.
orthogonal, that is b(M~,M~) = O.
Hence M is an orthogonal direct sum
g=~ I f 7~# ~" t h e n Mj-~= MT . i n M r l ~ ) M ~ ,
so t h e s e c o n d b i g summand i s n e u t r a l .
This splitting
i n d u c e s a map
of linking
pairings
~=~ Furthermore,
s i n c e R~ i s a d i s c r e t e
the Witt group of nonsingular kc./. = t~/~.. = R/~ o f R~. i n t h e f o l l o w i n g way. We(k~,-) + W (F,R~,-), an R - t o r s i o n
valuation
This is a standard There is a natural since
a vector
module and a n e n s i n g u l a r
R / R - ~ % o f K/R.
We(F,Rr
= W (k~,-),
E-Hermitian forms over the residue
s p a c e may be r e g a r d e d a s an ~ - l i n k i n g -1
ring,
'Chat i t
field
w h i c h may be proved
result,
monomorphism
s p a c e o v e r kq may be r e g a r d e d a s a-Hermitian pairing
is onto follows
f o r m on s u c h a v e c t o r
with values
i n t h e submodule
from the following
lemma
(also well-known) : Lemma 7
If b is a primitive
e-Hermitian
R-torsion
module M, and i f N i s a f i n i t e l y
t h a t N C I ~ t h e n (M,b) i s W i t t - e q u i v a l e n t primitive Proof
~-Hermitian pairing The i n d u c e d p a i r i n g
pairing
on a f i n i t e l y
generated to
generated
submodule o f M s u c h
(N'L/N,b N,
where bN i s t h e
i n d u c e d on N~--/N b y b .
b N on
N"~]N, d e f i n e d
bN(n 1 mod N, n 2 mod N) = b N ( n l , n 2) f o r a l l
ni,
by n 2 i n N~ i s c l e a r l y
a
132
primitive e-Hermitian pairing.
Let P = [ I P in N ~} be
the image of N ~ in M(~)(N~/N) via the diagonal embedding.
Then P is
selforthogonal with respect to the pairing b(~)(-bN) , for if b~(-b
N) (<m,
n mod N>, ) = 0 for all p in N , where
m is in M and n is in N ~, then b(m-n, p) = 0 for all p in N ~, and so m-n is is in P.
in N.
In particular m is in N ~ and <m, n mod N> = <m,m
Therefore
(M,b)(~(N~/N,-bN)
mod N>
is neutral.
//
Now i~ (M,b) is an E-linking pairing over a discrete valuation ring R~
and M is annihilated by m
submodule ~- mR/R of K/R. (for D(N,N) C ~2m-2(~-mR/R)
where m > l, then b takes values in the
Hence N =
~m-lMis such
a submodule of M
= O) and NZ/N is annihilated b y ~ m-l, so
on using the lemma repeatedly,
(M,b) is Witt-equivalent
to some (M",b')
where ~M' = O.
Clearly M' is then a vector space over k~, and b' takes
values in q l R / R
~k
Thus W (F,R,-) =
, so may be regarded as an e-Hermitian form on M'.
~_ W (k~,-) where k ~ =
R/~ and the summation is over
~=~ all primes of R left invariant under the involution. The ring A E is a localization of the ring Q(tl)Et2] of polynomials in one variable t 2 with coefficients
in the field Q(tl) , obtained by
inverting all polynomials
in t 2 with constant coefficients,
principal ideal domain.
It may also be described as the subring of
K = Q(tl,t2) generated by Q(t I) u Q(t2).
and is thus a
Thus a prime ideal in A E
is generated by an irreducible element P(tl,t 2) in Q[tl,t2] of positive degree in each variable, and those invariant under the involution correspond to such p for which also p(t~l,t~ |) = ~(tl)B(t2)p(tl,t 2) for some units ~(t I) in Q(tl) , ~(t2) in Q(t2).
In fact it follows easily from the
irreducibility of p and that it involves each variable,
that
mn ~(tl)~(t 2) = rtlt 2 for some r in Q and m,n in ~ ; furthermore since it
133
must also be true that P(tl, t 2) = ~(t[l)6(t~1)p(t[ I, t2 I) , r 2" must equal
I.
Thus p(tTl,t~ I) = • m n
The residue field of such
a prime ideal is an algebraic extension of Q(tl), k = Q(tl)[y ] say (where
y
satisfies P(tl,Y) = 0), and admits a non-trivial -1
generated by t I § t I
, y-~+ y
-1
-1
The elment ~ = t I - t I
involution satisfies
~ = -= and consequently k = k0[= ] where k 0 = {f in k I f = f} is the fixed field of the involution, by Galois theory.
Hence also if
6
is an e-Hermitian form on the k-vector space V, then ~B is an (-c)-Hermitian form on V, and so in discussing Witt groups of such fields, it suffices to consider s = +1, which anyway is the case relevant for the application to classical links. may be diagonalized,
Any (+1)-Hermitian form (V, B)
that is to say V splits as an orthogonal direct
sum of l-dimensional subspaces, and so there is an epimorphism = [k~/Nk/k0 k'] -~+ W+I (k,-)
[97~.
Not every prime ideal of AZ invariant under the involution is thus associated to the Alexander module of a boundary link; an 'integrality'
condition that must be satisfied.
L a ~-component boundary link, E (L) is principal,
there is
In general, for generated over A~ by
an element A~ such that e(A ) ~ A (I, ..., ]) equals 1 and A~ = A~. It follows that Eo(tHI(X') ~) is generated over A E by A . .-
i
is a t o r s i o n
module
over a
R
then
Now if -
and thus the primes occurring in the direct sum decomposition are just the divisors of Eo(M).
Assuming once again that ~ = 2, if AZ/~ e is a
direct s,~m~nd of tHl(X') ~ w i t h ~ p in A2.suchthat p = • lemma E
=~,
then~
is generated by some
and p(l, I) = I, for by the Gauss Content
] any factorization of A 2 into irreducibles in A~
comes from a factorization into irreducibles
in A 2.
134
On the other hand, if q in A 2 is such that q = q and q(l,l) = 1 then according to Guti~rrez there is a 2-component boundary link L with A2(L) = q [6~ . matrix
(~
(Alternatively,
Bailey's theorem implies that the
~] is a presentation matrix for H I(X') for some 2-component
link ~ ) .
In particular,
if q is irreducible,
then tHI(X') I = AE/(q)
is of length l, and so BE(L) maps to a nonzero element of W+l(k~.,-). Thus the image of the set of all 2 component boundary links in W+I(K2,AE,-)
(W+(kz,-) k
is contained in
I fl = ( p ) f o r
some p in A 2 such that p ( l , l ) =
I, %
p is irreducible and p = •
)
and maps non trivially to infinitely many factors of this direct sum, and so is not contained in any finitely generated subgroup.
A late addition:
Levine has announced work on the (unlocalized)
pairing on A = HI(X;A ) for ~ = 2 []20a].
Blanchfield
If ~(L) = I the kernel and cokernel
of the adjoint map from A to Hom (~,Q(t)/A) depend only on the linking number; if =(L) = 2 they are determined by longitude-annihilating bl(t2) and b2(tl) bl(]) = b2(1) = I, A/~A~
polynomials
(compare our Theorem VI.4) and an ideal I in A 2 such that bl(t 2) + b2(tl) - 1 is in I and e(1) = ~ .
I and, using Bailey's thesis, any such triple bl, b2, I
realized.
Moreover may be
135
Signatures
An argument similar to the one of the preceding section shows that (for ~ = I) there is an isomorphism W (JR(t), IRA, -) ~ ~ W where the sum is taken over all irreducible real polynomials
(~(t)/p(t),
-),
such that
(p(t)) = (p(t-])). Apart from t + 1 and t - I, which play no role in knot theory, any such polymomial must be a quadratic of the form pc(t) = t 2 - cos e. t + I, for some 0 ~ e $ ~. We(Q(t),A,
-) onto We(~(t)/pe(t),
signatures o 8 ~5,
132].
-) ~ ~
The induced maps of
are essentially the Milnor
Murasugi and Tristram have defined signatures
for any classical link, and have shown them to be concordance invariants D41,
193].
Certain of these signatures have been reinterpreted by
Kauffman and Taylor, and by Viro, who applied the G-index theorem to branched cyclic covers of D ~, branched over a properly embedded spanning surface for the link [88, 2 0 ~ .
A simple algorithm for the Murasugi
signature has been given by Gordon and Litherland
[603 .
The relation
between the Milnor signatures and the Tristram signatures of an Hermitean pairing have been elucidated by Matumoto
~26].
All of these signatures
appear to be related to (finite, branched) cyclic covers of the link. Cooper has defined "multisignatures"
for l-links analogous to the Tristram
signatures, but apparently more closely tied to the structure of the maximal abelian cover [34].
He observes that his invariants vanish for
slice links, but can be nonzero even if all the Tristram signatures are O.
Can all of these signatures be interpreted as homomorphisms
W+(K ,A~, -) to ~ ?
from
136
Appendix:
A Surgical View of the Blanchfield
In this section we shall describe
Pairing
the Blanchfield
l-knot by means of the surgery technique of Hempel and Rolfsen
[156].
characterize
[68], eevine
[1153,
In the latter papers this technique was used to
the Alexander polynomials
of a knot.
prove the theorem quoted in Chapter VII reworking of his argument,
[7].
Bailey used it to
Our construction
is a
paying closer attention to the Blanchfield
pairing;
indeed our ultimate
theorem,
and extend it to a characterization
of 2-component
pairing of a
goal is that we may better understand his of the Blanchfield pairing
links.
It is well known that a l-knot K may be unknotted by "replacing certain of the overcrossings
by undercrosslngs";
in the following lemma of Hempel as in Rolfsen
Lemma
(Hempel)
homeomorphism (i)
[157;
~8].
this idea is made precise
(We have refined the statement,
page 159] and Bailey
ET;
page 26]).
There is an embedding L : mS 1 x D 2 --+ X and an
h of U = S 3 - L(mS 1 x intD 2) onto itself such that
the induced map from mS I x {O} to S 3 is a trivial link, whose
.th l longitude and meridian are represented
by the images of
S 1 x {l} x {i} and {l} x S 1 x {i} respectively,
for l ~ i ~ m;
(ii) h maps each boundary component of U to itself and (on the i th meridian) (iii)
h(L(l,s,i))
= L(s,s,i)
for all s in S 1 and l ~ i ~ m;
each component of this link has linking number O with K and with
h(K); (iv)
hoK is unknotted
in S 3.
Let ~ : S l x S I ---+ ~X be an homeomorphism
carrying {I} x S I and
S I x {I} to a longitude and a meridian of K respectively,
and let
Y = X U~ S 1 x D s be the result of O-framed
Let T = mS I x D 2,
surgery on K.
137
V = -----~--~) X the pair
and W = V U~ S I x D 2 = y - T.
(U
U
T, K(SI))
g
(U U T, hoK(Sl)) Therefore
W U
g
is homeomorphic
T = (V U
of 0-framed
of the meridian
isomorphisms
identifies abelian
subspaces
spaces of these spaces
and lifts of maps)
of such covering unlabelledmaps
= 0.
surface,
(of. Chapter
HI(X'
U :R x D 2 ; ~ )
'
generated
the maximal coverings
of
The homology Al-modules.
groups
(All
of K lifts
is an isomorphism,
0 ---+ H2(Y,W;A)
while H2(Y;A)
(Y',W')
HI(W;A) = ~
Similarly
sequence
the boundary
is an isomorphism
free of rank m.
(: HI(~T;A)
HI(Y;A) =
= ~
sequence
A m must be null. homology
is null,
~
= 0,
----+ O
of equivariant to HI(~T;A)
=
= HI(T,~T;A)
gives a short exact
to HI(Y,W;A)
map from H2(Z;A)
from Ker
~
to HI(Y;A)
= H2(X;A) ~ Y Z
A m and HI(Y,W;A)
as the natural map from H2(Y;A) in the Mayer-Vietoris
X along a Seifert
map from HI(X;A)
of the pair D
by splitting
=
to a loop in X' which is null homologous
17 and so the natural
sequence
= ~ ,
A TM = H2(T,~T;A ) and HI(T,~T;A)
H2(Y,W;A ) = H2(T,~T;A ) =
so the long exact
therefore
a prime
T = Z
g
and hence
x S 2 = S 3 - S ~ , HI(Z;A ) = 0 and H2(Z;A)
Since X' may he constructed
there
(Z',W',T')
in Z.
are induced by inclusions).
the longitude
By excision,
(and the induced
by affixing
since T' = T x ~ , HI(T;A ) =
H2(T;A)
groups,
We shall denote
spaces are then finitely
Since Z' = ~ while
homology
T ~ S 3.
g
of K into each of X, Y, U, W and W U
each of these groups with ZZ .
covering
to Z = S 1 x S 2
of T and W (respectively)
on the first integral
=
on T.
surgery on the unknot K in U U
embeddlngs
Then
(S 3, hoK(Sl))
T) U~ S 1 x D 2 is homeomorphic
g
Let e and f be the induced
induces
to the pair
via the map given by h on U and by the identity
since it is the result
The inclusion
Let g = h -I o L I ~T.
for
so there
: HI(T;A) ) to HI(W;A) , which
is
138
As generators for HI(W;A) we may take the Alexander duals in Z' C S 3 of the images under e' of fixed lifts to T' of the cores of the components of T.
Thus if cl,...,c m are such lifts, whose images generate
HI(T;A) , the dual basis of HI(W;A) is determined by l-cycles al,...,a m such that ~(tmf~ai ' tn e,cj) , = I if m = n and i = j, = 0 otherwise 9 (Here s
denotes the linking number of two disjoint I-cycles
and 6 in $3).
The module H2(Y,W;A) is generated by the images under
L' of discs in T' transverse to the cores Cl,... , cm 9 such disc in Y', and let D(d i) = HI(W;A).
E n~
.th Let d.i be the i
~ r.. tna. be its image in l~jzm ljn J
Since D(di) is represented by ~d i, it follows that
ri3n. = link (f~ di, tncj), and since f', d i = g(l x S I • {i}) is homologous to e~c i in e'(T')) rij n = rji_n. D with respect to the bases
d. i
In other words if D and
a. , then D j
is the matrix of
= ~tr.
Furthermore
since the sequence above is exact (or since HI(Y;A) = HI(X;A) = tA(K) is a torsion module), 6 = det D
is nonzero.
We claim that the Blanchfield pairing on HI(Y;A) is given by ~(~uiai)
,
~(Evjaj)] =
~ t r ~ -I u
modulo A.
(Here (u i) and vj) are regarded as column vectors in A u and so the matrix product on the right lies in Q(t)). Let z be a l-cycle on W', and S a 2-chain on Y' such that ~S C W'.
Then
Iy,(Z, S) - Iw,(Z , S 0 W') = Is3(f~(z) , f~(S N W'))
since f' embeds W' into Z' C S 3.
This in turn equals
link (f~z, ~f~(S N W')) = link(f~z, f ~ S )
+ link (f~z, f', (S N ~W')).
Since 6HI(Y';A ) = O, there is a 2-chain S.on Y' such that 3S. = 6a.. J J J
139
Since ~aj is homologous to D
I
1~k~m
E homologous to I~k~m ( ~ - I ) kj ~ dK in W'. _ I E ~(ai), ~Caj)] - ~ n ~
l
Iy.
E E neZZ l~k~m
E n~
link(f~ ai' tn l~k~m E (6~)-]) kj f* ~ dk)tn modulo A
(6~~-I) kj link(f~ ai, tn f', ~ d ~ t n modulo A
modulo A
which establishes our claim.
and that 9
tn Sj)t n modulo A
(~m|)k j link(f~ ai, tn e~ Ck)tn modulo A E l~k~m
= (~)- l) .. Ij
the facts that n ~E
Therefore
E link(f~ tn 6 f' aj)t n neZg ai ' *
l l ~n~ZZ
=
1
d k , S 0 6W' is
(In simplifying the R.H.S. we have used
link (=, tn 6)t n is A-sesquilinear in ~ and 6,
is Hermitean).
CHAPTER X
NONORIENTABLE
SPANNING
SURFACES
In the course of studying the quadratic form on the total linking number infinte cyclic cover of a link complement, Murasugi defined the nullity of a link L, ~(L), to be 1 + nullity ( V + V tr) where V is the Seifert matrix for any connected spanning surface for L El38].
He showed that
(L) lay between I and ~, and was equal to the nullity of the Jacobian matrix evaluated at (-I,...,-l), and Kauffman and Taylor, in a rederivation of his results, showed that n(L) is invariant under I-equivalence
E88].
The Alexander nullity shares some of these properties, and is defined in a similar fashion.
In this chapter it shall be shown that, nevertheless, the
two invariants are distinct, and can take any values such that l .<~(L) .<~(L) .<~.
Our ex~nples are constructed from links spannable by
disjoint nonorientable surfaces, and we prove an analogue of ~mythe's theorem characterizing boundary links.
We show also that a ~-component
link which is so spannable must have Mmrasugi nullity ~, and that this condition is sufficient if ~ =2, and we give a geometric interpretation for the invariance of the Murasugi nullity under concordance, in the 2-component case.
Figure I
141
Definition
A ~-cc~ponent
there is an embedding P :
link L : ~S I + S 3 is a ZZ/22Z-boundary 1 [ U i § $3 1 .< i~<~
link if
of ~ disjoint surfaces U i such
that L i = P]~U i. The surfaces are not required definition of boundary figure
links.
to be orientable,
For instance,
in contrast
the link depicted
to the
in
1 is clearly spanned by two M~bius bands and so is a 2Z/27Z-boundary
link, but it is very far from being a boundary number 4.
As this link has ~ = I and n = 2, it already suffices
that the Alexander is the link 9612
and Muras~gi
nullities
for it has Alexander
link.
polynomial
There is a characterization
links in terms of the link group, analogous
(This
The link of figure V.I is a
spanning surfaces may be assumed orientable,
hemology boundary
to show
are not always equal.
in the tables of E|57~ ) .
more subtle example, disjoint
link, for it has linking
zero, and one of the yet is is not an of 77/2Zg-boundary
to that given for boundary
links by Smythe.
Theorem
l
A ~-component
link L is a 2Z/22Z-boundary
link if and only if
there is a map f : G + *~(Zg/22Z) which carries some i th meridian generator Proof U i.
of the i th factor of the free product,
Assume first that L is a 2Z/2ZZ-boundary
for each I .< i.< ~. link with spanning
Each such surface has an open regular neighbourhood
the total space of its normal bundle v.i in X. a disjoint
to the
Crushing
surfaces
homeomorphic
to
the complement
family of such neighbourhoods
wedge of Thorn spaces
E85; page 2 0 4 ] ,
are induced from the canonical
of
to a point collapses X onto the P X + V T(~i). The normal bundles ~. i=l I
line bundle qN over ~ p N
(for N large) by
classifying maps n i : U i § I~P N, and these maps induce a map
142
T(n)
: V T(v.) -~ V T(nN). l
homeomorphism
carrying
Now T(nN) is homeomorphic
to IRP N+|
the zero section to the hyperplane
E85; page 205] , and the inclusion of ~RP N+!
IRpN+I
(N+I)-conneeted map, and so the inclusion of V
which determines
is an
into
The composition
This map carries an i th meridian represented which meets U. transversally
at infinity
into IRP ~ = K(TZ/22Z,|)
P V IRP= = K ( * (ZZ/2Z~) ,I) is (N+ I)-connected. gives a map X + K ( * (2Z/2ZZ),I)
by a
of these maps
a map f : G -~ ~*(Z~/2Z~).
by a loop from the basepoint
in one point and is disjoint from the other
i
surfaces U. to the generator of the i th factor. For the map of spaces J l~pN+ l X + V carries such a meridianal curve onto the Thom space of_ the restriction
of ~N to a point,
the zero section hyperplane essential
in other words
to a curve which intersects
]RP N in one point, and hence which is
in T(~ N) = ]RP N+! , and so in ]RP ~ (since the inclusion
]RP N+I + ]IRP~ is 2-connected). Conversely,
given a map f : G + * 2Z/2ZZ , it may be realized by a
map F : X -> V I~P , since the latter space is an Eilenberg-MacLane K ( * (7z/2Zg) ,l).
By general position
V IRpN with N large. 9
i
th
2Z/27z factor,
identity
it may be assumed that f maps X to
If f maps an i th meridian
(since it con~nutes with this meridian,
of the corresponding moving F transverse
spanning
]RP N, and is transverse
to the
but lies in the commutator It may then be
to that hyperplane.
to the union of these hyperplanes I I IRpN-]
the inverse image F - I ( ~ the link. //
of the
curve into the hyperplane ~pN-|
be assumed disjoint from the sole singularity,
surfaces
longitude
of the other components).
assumed that F I~X i maps a longitudinal
from each other)
to the generator
then it must map the corresponding
subgroup modulo the meridians
space
the wedge point,
IRP N+I)
On (which may and hence
is a family of disjoint
143
In the original theorem of Smythe characterizing boundary links the spanning surfaces had trivial normal b~mdles, and the universal trivial line bundle IR (with base space a point) has Thorn space T(]R) = S I, which there played the role which T(~N) = ]RPN+! plays in the above theorem. A nonorientable surface is a Z~/2Z~-manlfold in the sense of E134].
A
similar application of transversality to high dimensional lens spaces shows that L has ~ disjoint spanning complexes, the i th being a 2Z/PiZgmanifold with no singularities on the boundary, if and only if there is a map G +
* (Zg/piZg) carrying some i th meridian to a generator of the l.
i th factor.
(For there is a 2N-dimensional Zg/p~-manifold in
LN(P) = S 2N+! / (2g/p2Z) whose homology class generates H2N(LN(P) ; 2Z/p 7z), the Poincar~ dual of H I(LN( p); Zg/p 2Z) E31; page 89] .
The relevance of
Zg/pZg-manifolds to knot theory was observed by Cooper.) Finite dimensional approximations IRP N to IRP~ have been used to facilitate the distinction between the base space (IRPN) and the Thorn space (]RP N+l ) of the universal line bundle. 2-dimensional complex, any N >~2 would
Since X has the homotopy type of a suffice.
(Note that similarly the
inclusion of LN(P) in L (p) = K(Z~/pZZ, l) is highly connected.) Smythe's characterization of homology boundary links suggests two possible definitions for a 2Z/2ZZ-homology boundary link.
Definition
(a) A ~-component link L is a ZZ/2ZZ-homology boundary link if
there are ~ disjoint surfaces (not necessarily orientable) U i in X(L) with ~U. C ~X(L) and such that ~U. is Zg/22Z-homologous to the i th longitude in i i
~X(L). (b) if
there
A ~ - c o m p o n e n t l i n k L i s a weak 7 z / 2 7 Z - h o m o l o g y b o u n d a r y l i n k
i s an e p i m o r p h i s m G § ~(Tz/22Z ) .
144
The above theorem then has the following
analogue
(with a similar
proof, which we shall not give).
Theorem 2
A link L is a 2Z/2Zg-homology
there is an epimorphism G + ~*(2Z/2ZZ) G + ,~(2Z/27Z) § ~(2Z/2Zg) 9
1
th
summand .
carries
boundary
link if and only if
such that the composition
the i th meridian
to the generator
of the
//
The last c l ~ s e
is superfluous
in the case of homology boundary
links, for any autemorphism
of 2Z~
F(~) = ~*2Z E123; page 168]
and so any epimorphism of G onto F(~) can be
changed so as to carry meridians
Theorem 3
can be lifted to an autemorphism
to standard generators
The Murasugi nullity of a ~-component
of
of the abelianization.
~/2~-homology
boundary
link is ~. Proof
Let L be a ~-component
link with group G.
number homomorphism from G to ZZ/2ZZ -I, determines
= {•
so A( * (Zg/2Zg))
isomorphism 2ZE~)(2Z/2ZZ ) ]
has a canonical A-module
= A/(t I + ! .... ,t + I).
and the map sending each
} induces the projection of
Hence if there is an epimorphism f : G § * (ZZ/22Z)
such that the composition with the map sending the generator to -| in { + I } is the total ZZ/2ZZ-linking epimorphism A(f) § 7z/27Z-homology (-l,...,-I) follows. //
to
= A/(tI2 -l ...... t 2 -l),
structure,
standard generator of (~(Zg/27z) to -I in { • (Tz/2Zg)] onto ~ .
linking
}, which sends each meridian
an epimorphism from A = ZZEG/G' ] to ~
There is a canonical
ZE
The total 2Z/2ZZ
: A(G) -~ A( , (Zg/2~)) * (Zg/2Zg))
=
of each factor
number homomorphism,
then the
gives rise to an epimorphism .
Since this is the case for a
boundary link, and since the Jacobian matrix evaluated at
is a presentation matrix for ~
(~A A(G) over ~ ,
the theorem
145
Corollary
Given integers
I .< ~ ~< q ~< p, there is a p-component
link with
Alexander nullity ~ and Murasugi nullity q. Proof
Let ~' = ~ - ~ +
I and q' = q - ~ +
;.
Let L o be a ~'-component
all of whose pairwise
linking numbers are odd, and let L' be the link
obtained by replacing
the i th component of L o by its (2,1)-cable
by the boundary of a M~bius band whose core is that component)
AI(L)(tI,I .... ,I) ~ 0.) components
The link obtained by deleting
of L' has Murasugi nullity
(that is,
for i .< i .< q'.
Then ~(L') = ; since all the linking numbers of L' are nonzero. follows from the second Torres conditions, used inductively
link
(This
to conclude that
the first
(n' - l)
I, so q(L') .< l + (n' -I) = q'.
On
!
the other harml the group of L' maps onto B, (ZZ/2ZZ) the theorem shows that q(L') >. q'. trivial
(~-l)-component
Therefore
q(L') = q'.
Let L" be a
link and let L = L' J _ i L" be the disjoint union
(so that L' and L" are separated by a 2-sphere q(L) = q and ~(L) = ~, since these invariants disjoint unions.
and so an argument as in
in S 3) .
Then ~(L) = a,
are clearly additive for such
(If ~ >. 2 we may construct an example more simply by
taking the disjoint union of an q'-component linking numbers nonzero,
a (U-n+
Zg/2Zg-boundary
l)-component
link L 1 with Hosokawa poly-
nomial V(L2) = 1 (so that q(L2) = I) and an (a-2)-component We may consider also the reduced ! .< ~(L) ~ K(L) .< n(L) ~< ~.
link L 1 with all
trivial link.) //
nullity and the sequence
Are the members
of this sequence independent?
The proof of the theorem actually shows that if L is a weak 7z/2Zg-homology
boundary
link and if the epimorphism f : G + ~* (Zg/22Z)
is
such that composition with the map sending the generator of each factor of * (Zg/27z)
to (-I) in {-+l } is the total Zg/22Z-linking number hemc9
of L, then q(L) = B. homology boundary
It is easily seen that if ~ = 2 such a weak 2Z/2~-
link is a ~E/22Z-homology boundary
link.
In their
146
discussion of the Murasugi coverings
nullity,
Kauffman and Taylor used branched
associated with the total 2Z/22Z-linking number homomorphism.
The boundary of an annulus embedded two full twists is a 2-component
in S B with unknotted
core and
link with linking number 2 and group
presented by { a,x] (ax) 2 = (xa) 2 } which maps onto ZZ * (2Z/22Z) = { u,v[v 2 = i} via the map sending a to u and x to u-lv,
and hence is a weak 2Z/2~Z-homology
boundary
below).
link (as follows
homology boundary
from Theorem 4
link and 2Z/27Z-boundary
link are probably distinct
also, if ~ >.3, although we know of no examples However boundary
there is the following
llnk with 2 components
to show this.
pleasant fact.
is a 2Z/22Z-boundary
from the algebraic characterizations
The notions of ZZ/2~-
A Zg/2Zg-homology
link.
This follows
of such links, and is a consequence
Theorems 3 and 4, but can be seen more directly by considering Seifert
surfaces meet the components
section through a neighbourhood
of ~X.
how the
In Figure 2 is shown a cross-
of the first component of sucha link, and
visibly all but one of the (odd number of) boundary surfaces parallel
of
components
to this component may be successively
desingularized.
Figure
2
of the Seifert
paired off and
147
Moreover, whether a 2-component
link is a Zg/22Z-boundary link is
determined by its ~lurasugi nullity.
Theorem 4
Let L be a 2-cemponent
(I)
n(L) = 2,
(2)
Al (e) (-l ,-I ) = 0,
(3)
L is a ZZ/22Z-boundary link.
Moreover
Proof
link.
Then the following are equivalent"
if these hold then the linking number of L is divisible by 4.
Let B denote the A2-module G'/G", and let ~
A2/(tl+ l,t2+ I).
(As a ring ~
denote the A2-algebra
is isomorphic to ~
= A2/(t I- I, t 2- I).)
Now El(L) = (AI(L))(t l-l,t 2 - I) by Theorem IV.2, while by lemma ]II.6 El(L) = E0(B).EI(1)
= E0(B)(t I -I,t 2 - I).
Since the Jacobian matrix of G
evaluated at (-1,-I) is a presentation matrix for the ~ - m o d u l e n(L) = 2 if and only if E I ( ~
O
QA
A(L)) = 0 if and only if AI(L)(-I,-I)
Assuming this is the case, then E 0 ( ~ (The kernel is the ~ - t o r s i o n
~
A(L), = 0.
~) B) = 0 and so ~ (~ B maps onto ~ .
submodule of
~ Q
B, for El(B) ~_ E2(L) by
Traldi's result of Chapter IV, and E2(ZZ (~ A(L)) is the unit ideal (I) of 2Z, so E2( ~ G
A(L)) must be nonzero, as these ideals have the same image
E2( (7z/2Z~) ~ A EI(~
A(L)) in the common quotient A2-algebra Z~/27z.
(~ B) fi 0.)
Hence there is a A2-epimorphism p : B + ~
Hence and ker p,
being a A2-submodule of B = G'/G", is a normal subgroup of G/G". r = (G/G")/ker p .
Let
Then there is a commutative diagram
!
--+
G'
--+
G
a --+
ZZ 2
-+
1
1
-.-+
~
--+
F
--+
~2
"-'+
1
where a carries an i th meridian x. to the i th standard generator of 2Z2 i
148
(for i = 1,2).
Therefore
F is generated by the images q(x I), q(x 2) and by a
generator of ~ ,
and the A2-module
structure on ~
conjugation with the images of the meridians.
= F' is given by
Therefore F has a
presentation {YI'Y2 'z [ Yl z Y f I = Y2 zY2 - 1 = z-l, E y l,y2 ~
= zk }
for some k, which clearly n~ast be odd since F/F' = 7z2 . meridians
x i carefully,
may assume k = [.
and on replacing
By choosing
the
z by its inverse if necessary, we
For suppose that E Y l , Y 2 ] = z4j-+l' and let w be an
element of G such that q(w) = z. x I' and x 2' are meridians
Let x I' = x I and x 2' = w j x 2 w -j .
such that E q(xl'), q(x2')]
generates
F'.
Then Thus F
has a presentation {Yl,Y2,Z
I z = Eyl,Y2],
where the generator Yi corresponds 9 th l meridian.
This presentation
Y] zyl-I
= y2 zy2 -I
to the element q(x i), the image of an is Tietze-equivalent
{YI'Y2 I Yl2 Y2 = Y2 y 2 ' Yl y22
carries an i th meridian Eg/2~-boundary
link.
to
= y 2 Yl }
and so the centre of F is generated by yl2 and y 2 . of F onto F/centre F ~
= z-I }
(2Z/22Z) * (2Z/22Z);
Let s be the projection
then s o q : G § (2E/22Z)* (2Z/22Z)
to the generator of the i th factor, and so L is a The implication
To simplify the notation,
(3) =>(I)
is contained
in Theorem 3.
let A(tl,t 2) stand for Al(L)(t],t2).
By
the second Torres condition A(t I,I) = (tl~ - I/t I - l)A I(L l)(t I) where = [A 1 (L) (I ,I) I is the absolute value of the linking number, and analogously for Al(l,t2).
Hence if A(-l,-I)
A(-l,|) = A(l,-l) = O. A(tl,t2)
=
= O, so A(l,l) is even, then
On expanding A(tl,t 2) about (-;,-l) as
A(-I,I) + a(t I + I) + b(t 2 + I) + (terms of order
it follows
that A(-],-I) = 0 implies that
>.2
in
t2§ 1 and t2+ I) ,
149
A(I ,-i)
0
0 + a.2 + b.O + (terms of order
~2
in t I + 1 = 2
and and hence A(1,1)
that a is even, and similarly b is even.
= 0 + a.2 + b.2 + (terms of order
is divisible
by 4.//
Corollary.
If a 2-component
a 2Z/22g-boundary
link.
It seems u n l i k e l y 7z/22Z-homology boundary
=0)
Hence
in t I + I = 2 and t 2 + I = 2)
link has Alexander
polynomial
zero,
then it is
// that in general link.
~(L) = ~ should imply that L be a
In the case ~ = 2, the group
D = (2Z/22Z)* (Tz/27z) is fortuitiously discussion
~2
t2 + 1
in terms of Alexander
metabelian,
and thus amenable
to
ideals.
An epimorphism G + D induces an epimorphism G2/G 3 -> D2/D 3 ~
77/277 of the
second stages of the lower central
series of G.
2-component weak homology boundary
link L, then the linking number of L is
even
E27].
(This also follows from M i l n o r ' s
second Torres condition
implies
= 0.
that L is a 2-component
homologous
analogous
that such a
can be g i v e n a m o r e geometric
proof.
link such that L 1 = ~M 1 and L 2 = ~M 2 where M 1
in S 3 and M 2 is a 2Z/qZZ-manifold
is the closed curve representing
characteristic
to that of
link.
in S 3 - M 1 .
to p.B(M I) in S 3 - M 2 while L 2 is homologous
(where B(Mi)
the
link L has even linking
An argument
The condition on the linking number
is a 7Z/pZZ-manifold
Conversely
= A 2/(t I - ],t 2 + l) instead of ~. then shows
link is a weak Zg/2Zg-homology boundary
Suppose
theorem.)
that a 2-component
number if and only if AI(L)(I,-I) T h e o r e m 4 with 77t
Hence if G is the group of a
to q.~(M 2) in S 3 - M I
the Bockstein
class of the singular m a n i f o l d M i E 1 3 4 ] ) .
linking number of L 1 and L 2 is divisible by pq.
Then L 1 is
of the Therefore
the
150
Definition
A concordance.~
: pS 1 x E0,1]
§ S 3 x E0,i]
between
L = ~f]~S I x { 0 } and L' = .~[pS I x { I } is a ~/22Z-boundary extends
to an embedding
~:
I I
which meet
~(S 3 x E 0 , I ] )
V. 0 ({ i } x S I x F 0 , 1 ] ) i V.' i
= ~(Wi)
Furthermore assumed
transversally U V.' i
N (S 3 x { I } )
and ~ P [ { i }
.~ is a boundary
carrying
links;
moreover
an i th meridian
5
map j : G - > >
~I(S 3 x EO,I]
to the generator
denote
induces
W. may be i
link.
the groups
theorem.
{u,v[u 2 = v 2 = I}.
- im.~)
n~
-~ D/Dn'
by t 2n-!, where (D/Dn)
if and only
~*(Zg/22Z)
from L to L',
Then .~ is a Zg/27Z-boundary link.
of L and -~respectively.
Jn : G/Gn §
J/>n
Let f : G + D be a map
has a
carrying meridians
Then f induces
family
and thus a map F : >
The natural
on all the nilpotent
The ~roup D = 7Z/22Z*2Z/22Z
and so there is a compatible
Fn = fn O Jn-I : J / ~ n
to
be a concordance
x,y for L to (the images of) u and v respectively. : G/G n § D/D
concordance
of the i th free factor.
-~ S 3 x E 0 , | ]
isomorphisms
by Stallings'
presentation
= I~
=
from L to L' then L and L' are each
and hence L' is also a 2z/22Z-boundary
Let G and >
quotients,
>=
L is a 7z/22Z-boundary
concordance,
generated
3-manifolds
= ~-f[{i} x S I x E0,I]
-~ is a 2Z/2%g-boundary
Let -~ : 2S 1 x ~ 0 , I ~
and suppose
n
~>(~Wi)
if all the 3-manifolds
concordance
if there is a map of the group
f
of disjoint
if
N (S 3 x { 0 }),
x S I x E0,i]
concordance
concordance
orientable.
Zg/2Zg-boundary
Proof
and such that
where V i = ~ ( W i )
If .i~is a ~ / 2 ~ - b o u n d a r y
Theorem
W. § S 3 x EO,I] i
.
1
links
epimorphisms
of epimorphisms
-~ lira§ (D/Dn) "
Now Dn Is
t is the image of uv in D, and ~ D n -~ { I }, so n>.l
is the completion
of D with respect
to the topology
for which
151
{D
n
} is a neighbourhood
basis at I.
This clearly induces the 2-adic
topology on the infinite cyclic normal subgroup of D generated by t, and so there is an exact sequence 1
--+
where the conjugation by multiplication
2Z 2
~
I)
by -| .
Therefore K = Zg2 N i m F
containing
i(t).
7Z/22Z
1
Let i : D § D denote the natural
inclusion.
The
normal subgroup of D which contains
i(D).
is a finitely generated Let w generate
torsion free abelian group
the maximal
i(t), and let K o be a complementary
is normal in i m F ,
--+
action of ZZ/22Z on the 2-adic integers 7z2 is generated
image of F is a finitely generated
which contains
~
cyclic subgroup of K
summand to 2Zw in K.
Then K o
and i m F / K o is generated by the images of u and w.
The
map ~ : D § i m F / K o sending u to i(u).K o and t to w o K o is an isomorphism, and so ~ = ~-I o F F(j(y))
:#
-~ D is an epimorphism.
= i(v) = i(u)i(t),
Since the abelianization
so ~(j(x))
of >
= u and ~(j(y))
__+
2Z2
B --+
(PZ/27Z) 2
where ~ maps j(x) to (l,0), j(y) to (0,l) and v to (0,i) and p is reduction modulo 2.
0 ker~
= ut k
Hence there is a commutative
D
J /ker~
= i(u) and
>'
square
to (0,0), 8 maps u to (I,0),
Therefore ker ~ / k e r ~ 0 ker ~ ~
and so as in the discussion
ker
of the group F in Theorem 4,
is generated by the images of j(x) and j(y).
generated by the images of j (x) and j (y) and so k m~st be _+l. may be changed by composing ~ with conjugation by u. that k = + !
for some k in ZZ.
is generated by the images of the meridians
j(x) and j(y), k must be odd.
is infinite cyclic,
Now F(j(x))
Therefore D is The sign of k
Thus it may he assumed
and so ~(j(y)) = v and the theorem is proved. //
152
The last assertion of the theorem is also a consequence invariance of Murasugi Theorem 4.
nullity under arbitrary
1-equivalence
does Theorem 5
It is not true in general be a boundary
concordance.
have a corresponding
4-discs
Then LI2S 2 - L-l(intB I
rank 2, as in the examples
links need
2-1ink with group G,
in S 4 which each meet each component of
in a 2-disc and are such that the disc links LIL-I(BI)
link to itself with group G~
extension?
that a concordance between boundary
For let L be a 2-component
and let B I and B 2 be disjoint
trivial.
[88~ , and
The result of Kauffman and Taylor applies to links with any
number of components;
imL
of the
and LIL-I(B 2) are
U i n t B 2 ) is a concordance
from the trivial
Now if G cannot map onto a free group of
of Chapter II, no concordance with group G can be
a boundary concordance. In higher dimensions
similar considerations
every 2-component n-link is a ~ / 2 ~ - b o u n d a r y Alexander
ideal must vanish,
In other applications Clark has considered spanning
as follows
apply.
n-link if n ~ 2, for its first
from Stallings'
of nonorientable
surfaces
the minimum number of crosscaps of any nonorientable
and Litherland have used nonorientable
conversation
Theorem and Theorem V.2.
to classical knot theory,
surface for a knot as an invariant of the knot
for the Murasugi
In particular
sDanning
signature of a link [607 .
that the analogous Tristrem-Viro
means of ~ / p ~ - m a n i f o l d s
spanning
geometric proof that (2-component)
the link.)
surfaces
[28],
while Gordon
in their algorithm
(Cooper has remarked
in
signatures may be studied by A final question:
slice links are
~/2~-boundary
is there a links?
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MR 81g:57005
INDEX
adjoint map of a pairing
Ad b
1]7
Alexander class
6(M)
119
Alexander ideals
Ek(L)
4O
Alexander module
A(G), A(L)
40
Alexander module, truncated
Aq(G)
59
Alexander nullity
=(L)
42
Alexander polynomial
Ak(L)
40
Alexander polynomial, reduced
Ared(L)
102 112
algebraic link annihilator ideal
Ann M
augmentation of the Laurent polynomial ring
e:A
27
§
1!
BAILEY: theorem on presentations of link modules (Bffi2)
93
Blanchfield duality
120
Blanchfield pairing
Bs(L), BE(L) BLANCHFIELD: theorem on symmetry of Alexander polynomials
121 84,
119
boundary concordance
124,
150
boundary concordance, ~/2 ~ -
150 7
boundary link boundary link,
t41
~/2 ~ -
12
Cartan-Leray spectral sequence Chen group
Q(G;q)
5O
Chen kernel
G(~)
63 43
COCHRAN: lemma on H2(G')
3
concordance
12
conjugate module
99
Conway identity
113
cover, finite cyclic
10
cover, maximal abelian
X'
cover, maximal free
X~
13
cover, total linking number
XT
101
Cramer's rule
28
CROWELL:equivalence of corresponding group and module sequences
4!
Crowell exact sequence
4O
CROWELL:theorem on splitting llnk module sequences
48
176
CROWELL and BROWN: theorem on homology boundary links (~=2)
67
CROWELL and STRAUSS: theorem on elementary ideals
49
divisorial hull
29
Ek(M)
27
equivariant (co)homology
H,(X;A)
11
exterior, link
X(L)
elementary ideal 1-equlvalence
3
4
fibred link
109 50
group, Chen
Q(G;q)
group, free
F(~)
group, free metabelian
F(~)/F(~)"
group, link
G(L)
4
group, ribbon
H(R)
2!
8
GUTIERREZ: characterization of trivial n-link (n ~ 4) HEMPEL:
8 136
lemma on unknotting by surgery
Hermitean pairing, (c-)
|]7
homology boundary link
8
homology boundary link, (weak) ~ / 2 ~ HOSOKAWA:
143
characterization of reduced Alexander polynomial
Hosokawa polynomial
V(L)
37
-:A
§ A
isotopy, local
11 3
KERVAIRE: characterization of n-link groups (n ~ 3) KIDWELL: theorem on divisibility of reduced Alexander polynomial knot
5 ]O6 2
Laurent polynomial ring LEVINE:
103 102
ideal class involution of the Laurent polynomial ring
67
A
32
lemma on elementary ideals L:~S n -> Sn+2
link
11
link, boundary
2 7
link exterior
X(L)
4
link group
G(L)
4
link,
homology boundary
link-homotopy
8 4
link module (Bailey)
93
llnk module sequence (Crowell)
48
link, ribbon
16
link type
2
177
6
linking number
118
linking pairing
3
local isotopy longitude, longitudinal
6
curve
MASSEY: theorem on completions of llnk module sequences
51
Mayer-Vietoris sequence for (homology) boundary link
14, 5
meridian, meridianal curve
135
Milnor signature MILNOR: theorem on presentation of nilpotent quotients
10 112
monodromy Murasugi nullity
n(L)
140
MURASUGI: theorem on Chen groups (B=2)
58
MURASUGI: theorem on link-homotopy
98 118
neutral linking pairing
79
NIELSEN: theorem on primitive elements in F(2)
I0
nilpotent quotient
G/G
normal closure
<<S>>
n
5 3
null concordant nullity, Alexander
a(L)
42
nullity, Murasugi
B(L)
140
nullity, reduced
K(L)
108 2
orientation convention
117
perfect pairing
12
Poincarg duality preahelian presentation
10
primitive element of free group
79 117
primitive pairing Property R
18
pseudozero
32 27
rank reduced Alexander polynomial
Ared(L)
102
reduced nullity
K(L)
108 17
ribbon conjecture ribbon group
H(R)
21
ribbon link
16
ribbon map
16
ROLFSEN:
theorem on isotopy of links
3
12
178
SEIFERT:characterization of knot polynomial
91
Seifert surface Seifert surface,
7 singular
8
sesquilinear pairing
I17
short free resolution
27
slice link
3
slit
16
SMYTHE: conjecture on links with Alexander polynomial 0
54
SMYTHE: theorem on boundary links
8
splittable link
5
STALLINGS: theorem on homology and nilpotent quotients Steinitz-Fox-Smythe invariant
0(M), y(M)
sum of linking pairings
17, 136
throughcut
total linking number cover
37 I18
surgery
torsion (sub-) module
I0
16 tM XT
total linking number homomorphism (~/2 ~ )
27 lOl
144
Torres conditions
83
TRALDI: theorem on E _l(L) modulo 12B
90
TRALDI: theorem on higher elementary ideals in link module sequences 49 TRALDI: theorem on higher elementary ideals of sublinks Tristram signatures
86 135
Universal Coefficient spectral sequence
12
Wang sequence
lOl
weak concordance
116
Wirtinger presentation
9 118
Witt equivalence Witt group
W (F,R,-)
ll8
