On Some Properties of One-Valued Transformations of Manifolds

MA THEMATICS: H. HOPP

20fB

Pitoc. N. A. S.

Moore, R. L., Trans. Amer. Math. Soc., 17, 1916 (131-164). Hereafter we will refer to this paper as "Foundations." 10 "Foundations," theorem 6. l1 Ibid., theorem 28. 12 Ibid., theorem 29. lt If K is a closed subset of M, M(K) is a continuous curve. See "Arc-Curves," theorem 7. 14 "Arc-Curves," theorem 8. 16 If P is a cut-point of M(K), every component of M(K)4P contains at least one point of K. See "Arc-Curves," theorem 9, part (1). 16 Whyburn, G. T., these PROCIEDINGS, 13, 1927 (31-38), theorem 10. 17 Moore, R. L., Math. Zeit., 15, 1922 (255). - "Arc-Curves," theorem 10, part (5). 19 Schoenflies, A.,,Die Enticklung der Lehre von den Punktmannigfaltigkeiten, Zweite Teil, Leipzig, 1908 (237). 20 Moore, R. L., these PROCEEDINGS, 11, 1925 (469-476), footnote 5. 21 See an abstract of a paper by R. G. Lubben, "The Separation of Mutually Separated Subsets of a Continuum by Curves," Bull. Amer. Math. Soc., 32, 1926 (114). 22 Subscripts are reduced modulo n.

ON SOME PROPERTIES OF ONE-VALUED TRANSFORMATIONS OF MANIFOLDS By H. HoPp DEPARTMENT oP MATHEMATICS, PRINCETON UNIVERSITY

Communicated February 11, 1928

1. Some Constants of a Transformation.-Given any set of v-cycles in an n-complex A', its maximum number of independent cycles with respect to homologies shall be called the "rank" of the set; it is not greater than the vth Betti number 7r, of U&. Consider another n-complex M" transformed into ii" by a one-valued continuous transformation f and the set of all f(c), where the c are the v-cycles of A; then the rank r, of this *set is a constant of f (v = 0, 1, ..., n). If c;, C, ... c, where p, is the vth Betti number of M", form a fundamental set in MA and y'o, y'2 ..., 'yi,r a fundamental set in ML, then the. transformations of the v-cycles define a system of homologies'

f(c;)

k-i

y' (j = 1, 2, . . aJ,ek

PP)

(1)

and r,, is the rank of the matrix

A,= Ia,kII. When the c; and Y' are replaced by other fundamental sets, A, is trans-

MA THEMA TICS: H. HOPF

VOL. 14, 1928

207

formed by square matrices whose determinants are equal to 1. Therefore, also the invariant factors of the matrices A. are constants of f. If P. = ir,, for instance in the important case where A' is identical with M", the A, are square matrices and their determinantsr =

|A'

Y'.py ak (j, k = 1,'2, a,,= (whose absolute values are the products of the invariant factors), are also constants of f, to be considered. In this paper we shall deal with geometrical properties of f which are expressible in terms of the constants r, and a,. We shall assume that Mn and ,un are closed connected manifolds. Therefore,

Ao= 1111, Ali= IaxII,

Po=Pn-7ro=7r=l,

where the constant a. = a is the Brouwer degree2 of f. 2. A Formula of Contragredience.-Let L= | (ci.c ') || Av = ('y!.'y'') || be the intersection matrices of the fundamental sets c!, c. -' in Mx and Y,, 'Jr' in IA' respectively. Their determinants are *= 1,8 hence the inverses Lv ', A, -1 are defined. We shall denote by Em the matrix unity of order m, and as usual by B' the transverse of a matrix B. We shall prove below (Nos. 6, 7) the following relation of contragredience I

(-X

= aE,,

(2)

from which all our results will follow at once. 3. Properties of the Constants r,.-Let first be a # 0. Then aE,,, is of rank 7r,; therefore, the rank of no matrix on the left hand of (2) can be 0, with a degree different from zero. Let us now consider the case a = 0. Since the determinants of Ln-1, and A-', are # 0, the matrix (L1, An_.. A,-,)' =B = bij has the rank rn,. Hence the system P,

E bijx=1

=

O (i

=

1, 2,

7r)

a20

MA THEMA TICS: H. HOPF

PROc. N. A. S.

has 7r,-r,-, linearly independent solutions xl,", x2m, ..., xp,,m(m = 1, 2, 7r,- rn,). Since a = 0, according to (2) the elements c4k of A, satisfy the equations Z bijc4k = 0

(k = 1,2, .. .,i,).

.a . ., a;k of A there are at most 7r,-r"_ Hence, among the columns a'k, l2k, linearly independent; therefore, THZORJM II. Iff is of degree 0 thenr, + re., _7r, (v = 0, 1, ..., n). From now on we shall always assume that Mn and j?' are identical, although our results will also hold when merely P. = 7r,, even if Ml' pu. The sets of all v-cycles of M" and of all (n - v)-cycles have the common rank p,, so that the sum of their ranks is 2pv, 27r,. Therefore, theorems I and II may now be interpreted in the following manner: If the transformation f of Mn into itself is of degree $ 0 then there exist no c' nor c -, not 0 with an f-transform ~ 0. On ihe other hand, when f is of degree zero then for each v there exist v or (n - v)-cycles not ~ 0, but whose ftransforms are ~ 0. The sum of the ranks of these "degenerating" v- and (n - v)-cycles is at least half the sum of the ranks of all v- and (n - v)-cycles. 4. A Relation between the Constants a,; an Application.-The determinants of the matrices L and A are always 1; under the assumption == 1, becauseL=L. M = ,n we have even | L 1,I = A,., Therefore, by computing the determinants in (2) we find: THIOREM III. The constants a, and the degree a of a transformation of M" into itself are related by the equations a... a. = a' (v = O, 1, ..., n). We give an example of an application of this theorem: let Mn be the complex projective plane P. It is a 4-dimensional closed orientable manifold with the Betti numbers

IA"_

, p2 PO= P4 = 1, PI = P3 =O Therefore, III gives a2 = a, so that we have THSORP,M IVa. The degree of any transformation of the complex projective plane into itself is a perfect square. An example of a transformation with the degree b2 with arbitrary b is given by z' = Z (i = 1, 2, 3), where z: z2: z3 represents a point of P and z:z: 'its image. Furthermore, according to the formula of Lefschetz4 the algebraic number of fixed points under a transformation of P is 1 + a2 + a4; but now we see that this number is equal to 1 + a2 + a 2 $ 0 whatever the integer a2. Therefore, in analogy with a well-known property of the real projective plane,

VOL: 14, 1928

VMA THEMA TICS: H. HOPF

-

2

THZOR1M IVb. Every transformnation of the complex projective plane has at least one fixed point. 5. The Behavior of the Kronecker-Indices under a Transformation.-In the case Mn= u' (2) may be written

(L- t, Ax-, Ln-P)' A,

aEp,

(2')

or, since Ln,_ = (-1)v(+i)Lt and (BC)' = C'B',

(2") L,A'_LJT1A, = aEp,. When a 96 0 then also A, 0 0 (Th. I). Hence A,1 exists and from (2') follows L4A'_VL = aEp,A- 1. Furthermore, since EpJ and A, are commutative, LA'_,L7' = A,, .aEp, = (3) ,LA'-V-aL,,. Consider now the transformations of fundamental sets c!, by (1): 'f(cV) = Cv E I cvc k-i P', f(cj') = CJ _V ailC-

cj' given (1')

1=1

and the intersection matrix of the transformed cycles. L = || (cQ .c>') If| We have from (1') (Cs,.CV)

=

ia.k(ck k,1

acL)afl

hence,

LI =A,LAA'_ and in view of (3) the equation

L, = aL,, from which follows (c!.c>"ny) = a(c!.cj7') and generally

(5)

(c'.c' ') = a(c' c')(5 for arbitrary cycles- c', c"~ . Therefore, we have proved THsORuM V. When M undergoes a one-valued transformation of degree $ 0, all Kronecker-indices of cycle-pairs are multiplied by the degree. As a corollary of this theorem we have the fact that the property of a pair of cycles c', c' -, to intersect or not to intersect one another in the algebraic. sense is not only invariant under homeomorphisms of M' but under all transformations of M' into itself, which have a degree $ 0.

210

ioc. N. A. S.

MATHEMA TICS: H. HOPF

We may point out that the assumption a P 0 which we have used in deriving (3) and, therefore, in proving V is necessary; for a simple transformation of a surface of genus 2 shows that the statements of theorem V and of its corollary are not correct in the case a = 0. 6. The Method of the Product Manifold, introduced by Lefschetz, will be used in proving the contragredience formula (2). It may be described as follows:5 If x and t are points of Mn and 1j', respectively, and X = x X t the point of the product Mn X which represents the pair x, t, then we write

=fII(X)

x = P(X),

[X = x X

t]

(6)

and call P and H the "projections" on M' and j.A". We consider an ncycle rF in Mn X /A# which generally is singular and say that x and t are corresponding with respect to r1 if there exists an X on r,, so that (6) hold. If we interpret t as the image of x then this correspondence defines a "transformation" T of Mn into u"n, which may be symbolically expressed by T(x) = IIP-1(x) (7a) and similarly a transformation T-1 of

7-1(t)

y" into M", the "inverse" of T:

= P1-1

(t).

(7b)

T and T-1 are both, in general, multiply valued. The following construction gives the image T(c') of a v-cycle cv of Mn: the "cylinder" c' X iA' erected on cP in Mn X M"' intersects r" in a v-cycle r C' XI(h and the projection of this cycle on it' is the image of c':

T(c')

=

ii(rn.ce

jU#).

(8a)

T-'(-y)

=

P(rn.Mn x a')

(8b)

X

Similarly, we determine for each cycle 'y' of JA". Concerning fundamental sets of cycles in Mn X Au we have the theorem' that, if cX, oYjX (, = O, 1, .I. n; i = 1, 2,9 .. px; j = 1, 2, ... ., 7xr), are fundamental sets in M' and ju2, the set of all products c' >X('4 (X = 0, 1, ..., v) forms a fundamental set of v-cycles in M' X u'. The projection of A' c4 X 'y4 on Mn

P(A') = P(4' X ' ) = 4c' is, considered as v-cycle in M', _ 0, if X < v; likewise is

fl(A')

=

H1(

X

'4;)

=

MA, TE EMATICS: H.-OPFF

Voi, 14, .19

O on ,

r

211

if X > 0.- Thus, if we ha-ve any v-cycle

S E 1-il j.c' x l, (i = 1,2, .., pA; j = 1, 2, ..,X) (9I)

X-o i,

then Pr

P(rv)

[on MT]

(9a)

w7A-fj; [on,4"].

(9b)

i

i-i

ll(J')~

E

cc

Now let the cycle r, which defines the transformation T be

Sj.

E

rn~

x

c

aJ

(10)

X 0 i.j

In calculating T(c') and T-1("yk) by (10), (8a), (8b) we shall make use of 7 ( P)(f-S) (c (c X yS) [on MA X A"]. (11) (C" X r CQ x ',S) ( )(-f

From (10), (11) and (yn-\ sn) =

we find

n

rn. cp X

/Ax

(\ () x lyl tJ J-0

-0 i,j

.C x j

*,E -0 E i,j ), (4t CD) X -Yw In view of (c4.4c) = 0, if X + v < n, and of (9b), for II (r . ck X ,) the only terms that are essential are those in wvhich (-yj7.juA') is v-dimensional, i.e. X = n - v; hence,

T(c`)

H(rn.Ck

X

A'S)

e-

V. .C (d -CD).i

(12a)

Therefore, if T(c`) -Z j-i

a"Vk^

(13a)

then we get from (12a) Av

aIV =

e''(CD'.)

or in terms of matrices, with e,, = A, = L'.. en-v

Similarly, let be for T-1 ., .

T

ji el"Iif,

A, = j aj

= ()(1+1)L,vex

8

I (14a)

-'-yv)pk

T-1(z)~

-il

-,

PlRoc.,N. A. S.

MA THEMA TICS: H. HOPF

212

Then by means of (10), (11) and in view of (cx.M') = c: it

r'3.M'n x 'w~ E E

X=0 ij

(ij

EZ E'j.4Ct X 'Yj X=O ii (-l(n

) (n

ek

.M' X

')Ck X ('Y,n-x

'

According to (Y-X. y') = 0 for X > v and to (9a) we are only interested in those terms on the right hand where c4 is of dimensionality v, i.e., X = v; hence, T-'(y))= P (rnJMn X jy) ~Et,ke{j(-1)'3 ('y7'.'4)CS (12b) and thus from (13b) Irp

Bv= (-1)'f3' Z ^(n-

i.e., if B,=

v

j=1

kI3iI

(-1)'3'An,ec

B =

(14b)

Now one sees that between the matrices A and B which define the cycle transformations under T and T-1 there must hold a certain relation: On replacing v by n - v in (14a) we find e, = L1-iA-p, and since Ln-=

Therefore, from (14b) B= (- 1)n(+')AA' 1A'nL( = (-1)'+l) (L Let now a cycle y' of g' be transformed into back into T(c') = TT-'(y'4) = '4 We have

TT

'(^'k)

with a square matrix U, =

=

uj

(y= T(c

show that

'Y'k

j=1

cp

-'A =

-

* (15)

T-'('y') and then

u kj4 Uk

Then the homologies pp c' i=l

E#ZjT(c =Efl=jaV zv i

)

i,j

B,AI= U,,. Hence the matrices U, belonging to the transformation TT-1 of

A'" into

MA THEMA TICS: H. HOPF

VOL,. 14, 1928

213

itself are according to (15) expressible by the matrices A in the following manner: (16) U, -)(i)(n P"A_,'V

7. Proof of the Contragredience Formula (2).-Up to this point we did not separate' one-valued and multiply-valued transformations T. The introduction of TT-1 marks the place where they naturally part. If we go from y" to T-'(y`) = c' and from cv back to T(ce) = TT-1('y') = , then in the general case we do not return to 'y'. For, although the cylinder c' X 2in has in common with rM the cycle r.Ml X y', of which the projection on ;s is y', this cycle is not the whole intersection rF . c X js" and, therefore, the projection ii((r1. cv X gu") is different from y'. When, on the contrary, T is one-valued, then for each point x of Mn the product x X is" has only the point X = x X T(x) in common with r"; therefore, for each cycle A, of rF the cylinder P(A') X gn intersects r" only in the points of A'; hence, (17) r".P(AV) X , = A .

If we take A' = r".M, X 7', then from (8b) and (17) follows

r"..T-1(75') x An

rt.Mn X ,lp

=

(18)

and from (8a)

TT-1(7') = (IIM(r.Mn X 7y)

(19)

for each cycle y' of 1A. The right hand of this equation may be determined by the method which has yielded (12a); from (10), (11) and (c'x. M") = cX follows n MnXxW, 4i\.j(--Y')(nX-P)kX(y

rn

E

EV

For the projection II we may omit all terms ('yr". 7') with dimensionality 3 v, i.e., with X F 0; hence,

II(rn.Mn

X

-Y,,) _,E eu,j ir)(n-wz

I

But because p,, = 7ro = 1, the matrix EO: has only one element eo; from (14a) follows e0 = a., where a,, is the only element of the matrix A,, and this element is by definition.(cf. Nr. 1), the degree a of T. Hence,

ii(rn.fo Xmy1) 9) and from (19)

TT

(_1)n(P+1)ay,

*,%.I--(- 1)"(P+l)ayv.

(20a)

(20b)

214

MA THEMA TICS: G. A. MILLER

PROC. N..A. S.

Therefore, the matrix U, defined in Nr. 6 has been determined under the assumption that T is one-valued:

UV =

(-1

+ 1)aE,

(21)

where' E*, is the matrix unity. From (21) and (16) formula (2) follows immediately. 1 Cf. S. Lefschetz, (a) Trans. Am. Math. Soc., 28, pp. 1-49; (b) Trans. Am. Math. Soc., 29, pp. 429-462; particularly p. 32 of (a). The sign"" introduced in (a) means "" mod. zero-divisors. The fundamental sets of this paper are all with respect to the operation ~. 2 L. E. Y. Brouwer, Mathem. At&nalen, 71, pp. 92-115. 30. Veblen, Trans. Am. Math. Soc., 25, pp. 540-550. 4 (a), formula 1.1; (b), formula 10.5. 6 A great deal of Nr. 6 is only a summarizing report on facts which are included in the papers of Lefschetz, quoted above. 6 Lefschetz, (a) No. 52. 7 Lefschetz, (a) No. 55. The proof of this formula, not explicitly given there, can be obtained easily by the same considerations as for the formulas of (a) Nos. 53, 54. 8 Lefschetz, (b) 9.2.

HARMONY AS A PRINCIPLE OF MATHEMATICAL DEVELOPMENT By G. A. MILLER DZPARTMINT OF MATHZMATICS, UNIVERSITY OF ILLINOIS

Communicated January 27, 1928

In the preface of volume 6 of the "Collected Works" of Sophus Lie the editor remarked that these works constitute an artistic contribution which is entirely comparable with that of Beethoven. In both cases a few apparently trivial motives dominated the entire creation. In the case of Lie it was mainly the explicit use of the concept of group in domains where its dominance had not been explicitly recognized before his time. This concept was practically refused by the mathematical builders up to the beginning of the nineteenth century but became during this century "the head stone of the corner" largely through the work of Sophus Lie. While the work of Beethoven has reached and probably will continue to reach a much larger number of people than that of Lie, it is questionable whether those to whom Lie's work is actually revealed receive less inspiration therefrom or are less impressed by the marvelous new harmonies which it introduces into a wide range of mathematical developments. As Sophus Lie was a Foreign Associate of this Academy and his work

600

ERRA TA

tional and educational resources of the Grand Canyon. The research and investigation conducted under the auspices of the Academy is supported by a grant from the Carnegie Corporation of New York. The fossil remains discovered in the Unkar are the oldest yet brought to light in the southwest and, notwithstanding the fortunately relatively slight degree of metamorphism of the sediments, may be as old as the lower Huronian. On the other hand, they may not be older than the Keweenawan. It is probable that some at least of the types of organic deposits in the Unkar which are shown to have been laid down in very shallow waters disturbed by winds and currents, sometimes beneath rainy skies, sometimes under a hot sun, are due to algae of the blue-green family, the Cyanophyceae. The abundance and indicated variety of forms of deposit, some of which may be of animal origin point encouragingly to the possibility of greatly adding to our knowledge of the certainly highly differentiated late Proterozoic life, which may have had its beginnings as long in advance of the Paleozoic as from the beginning of the Paleozoic to the present. They also give promise of paleontological criteria characteristic of the different groups of sedimentary rocks and far more valuable for the time classification and correlation of the not too far altered pre-Cambrian formations than any means now available.

ERRATA TO THZ NoTE: "On Some Properties of One-Valued Transformations of Manifolds," these PROCEEDINGS, 14, 206-214, 1928. P. 208, lines 1, 2, 5, 7, read p instead of xr; p. 209, line 8, read IA,I instead of A,; p. 211, lines 20, 21, read e instead of e; p. 211, last line, the homology should be numbered (13b); p. 212, line 9, read , instead of B; p. 214, line 10, read J. instead of Y.; p. 214, line 12, read 71.1 instead of 1.1.