Borsuk-Ulam Theorem for Maps from a Sphere to a Generalized Manifold

Geometriae Dedicata 107: 101–110, 2004. # 2004 Kluwer Academic Publishers. Printed in the Netherlands.

101

A Borsuk–Ulam Theorem for Maps from a Sphere to a Generalized Manifold CARLOS BIASI, DENISE DE MATTOS and EDIVALDO L. DOS SANTOS Universidade de Sa˜o Paulo, Departamento de Matema´tica-ICMC, Caixa Postal 668, 13560-970, Sa˜o Carlos SP, Brazil. e-mail: {biasi, deniseml, edivaldo}@icmc.usp.br (Received: 7 May 2003; accepted in final form: 30 December 2003) Abstract. H. J. Munkholm obtained a generalization for topological manifolds of the famous Borsuk–Ulam type theorem proved by Conner and Floyd. The purpose of this paper is to prove a version of Conner and Floyd’s theorem for generalized manifolds. Mathematics Subject Classifications (2000). 57P99, 55M20. Key words. Borsuk–Ulam theorem, generalized manifolds, Stiefel–Whitney classes.

1. Introduction The classical Borsuk–Ulam theorem says that for every map f: S n ! Rk , there exists a point x 2 S n such that f ðxÞ ¼ f ðxÞ, if k 4 n. The famous version of the Borsuk– Ulam theorem proved by Conner and Floyd in [1], replace the Euclidean k-space Rk by a differentiable k-manifold M k . In the Conner and Floyd’s proof, the differentiable structure of the k-manifold M k allowed to make use of the Stiefel–Whitney classes. Munkholm showed in [2] that all differentiability hypotheses can be eliminated, if M k is assumed to be compact, replacing the differentiable manifold by a closed topological manifold. Since these manifolds have a local differentiable structure, it was possible to prove the result by using such classes. It has recently been defined in [3] the Stiefel–Whitney classes for a generalized manifold and this is a main point concerning our generalization: by following the same steps of Conner and Floyd we prove the theorem by using such classes. We will specifically prove the following theorem: THEOREM 1.1. Let f: S n ! M k a map from the n-sphere S n to a generalized manifold M k ; let Að f Þ ¼ fx 2 S n : f ðxÞ ¼ f ðxÞg. Then ðiÞ if n > k then dim Að f Þ 5 n  k; ðiiÞ if n ¼ k and f : H n ðM n ; Z2 Þ ! H n ðS n ; Z2 Þ is zero then Að f Þ 6¼ . Throughout the paper, homology and cohomology groups will always have  ech cohomology in the sense of [13, coefficients in Z2 ; H denotes Alexander–C c Chapter 6, Section 1], H and Hc denotes the singular homology and cohomology

102

CARLOS BIASI ET AL.

with closed support in the sense of [13, Chapter 6 Section 3]. By dim we understand the usual topological dimension and the symbol ‘ffi’ denotes an appropriate isomorphism between algebraic objects.

2. Preliminaries Generalized manifolds of dimension m which we consider in this paper are certain topological spaces M such that M is an ENR and for every point x 2 M; H ðM; M  fxg; Z2 Þ is isomorphic to H ðRm ; Rm  f0g; Z2 Þ (for details see [3, 9] and [15]). Such manifolds have recently been studied; for example, it has been known that there exist generalized manifolds which are not homotopy equivalent to topological manifolds (see [8] and [9]). In [4] and [5], it has been shown that the Poincare´ duality holds for locally orientable generalized manifolds. Furthermore, in [6], every generalized manifold has been shown to be locally orientable. Thus the Poincare´ duality holds for all generalized manifolds (for details see [7]). More precisely, if M is a connected generalized manifold of dimension m, then Hmc ðM Þ is isomorphic to Z2 and the homomorphisms c _ ½M : H i ðM Þ ! Hmi ðM Þ;

_ ½M : Hci ðM Þ ! Hmi ðM Þ are isomorphism for all i, where [M] is the generator of Hmc ðM Þ and is called the fundamental class of M (see also [3, 2.3, p. 278]). We call the above isomorphisms the Poincare´ duality isomorphisms and denote them by DM . Now, we recall the concepts developed in [3]: let M a generalized manifold, by using the Poincare´ duality isomorphisms together with the Steenrood squaring operation, it is possible to define the Wu classes and hence the Stiefel–Whitney classes wq ðM Þ 2 Hq ðM Þ by virtue of the Wu formula (for details, see [3] and [12]). This means that these classes can be studied from an algebraic topological viewpoint, so that the differentiable hypothesis is not necessary. Remark 2:1: If f: M ! N is a homeomorphism between generalized manifolds, then f ðwðN ÞÞ ¼ wðM Þ. In fact, let vðM Þ denote the total Wu class of M. Then we have hx ^ vðM Þ; ½M i ¼ hSqðxÞ; ½M i for all x 2 Hc ðM Þ. If x ¼ f ð yÞ, the right-hand side of (2.1) is equal to hSqðxÞ; ½M i ¼ hSq f ð yÞ; ½M i ¼ hf Sq ð yÞ; ½M i ¼ hSqð yÞ; f ½M i ¼ hSqð yÞ; ½N i ¼ hy ^ vðN Þ; ½N i

ð2:1Þ

103

A BORSUK–ULAM THEOREM FOR MAPS

¼ hy ^ vðN Þ; f ½M i ¼ hf ð y ^ vðN ÞÞ; ½M i ¼ hx ^ f vðN Þ; ½M i which implies that wðM Þ ¼ f wðN Þ.

vðM Þ ¼ f vðN Þ.

Then,

by

Wu’s

formula,

we

have

Remark 2:2: Let M and N generalized manifolds of dimension m and n respectively such that k ¼ n  m > 0 and f: M ! N a proper continuous map. Let us denote by jk 2 H k ðN Þ the Poincare´ dual of f ½M 2 Hcm ðN Þ, in other words, f ½M ¼ jk _ ½N . Note that f ½M 2 Hcm ðN Þ is well-defined, since f is a proper map (see [14, p. 118]). Let the total Stiefel–Whitney classes of M and N be denoted by wðM Þ 2 H ðM Þ and wðN Þ 2 H ðN Þ respectively and let w ðM Þ denote the dual Stiefel–Whitney class of M,i.e., w ðM Þ ¼ wðM Þ1 . Define wð f Þ ¼ f wðN Þ ^ w ðM Þ

ð2:2Þ

which is called the total Stiefel–Whitney class of the stable normal bundle of f and we denote wk ð f Þ 2 H k ðM Þ the degree k term of wð f Þ, which is the kth Stiefel–Whitney class of the stable normal bundle (see [3]). In [3] the following class was also defined yð f Þ ¼ ð f jk  wk ð f ÞÞ _ ½M

ð2:3Þ

and the following results were proved (see [3, Lemma 2.6; Corollary 3.5]). THEOREM 2.3. Let V be an open subset? of N which contains f ðM Þ and consider the c ðM Þ, and consequently map fV ¼ f: M ! Vð N Þ. Then yð f Þ ¼ yð fV Þ in Hmk wð f Þ ¼ wð fV Þ. THEOREM 2.4. Let f: M ! N be a proper topological embedding of a m-dimensional generalized manifold M into a ðm þ kÞ-dimensional generalized manifold N, with c k > 0. Then yð f Þ 2 H mk ðM Þ vanishes, and consequently wk ð f Þ ¼ f ðjk Þ. c ðM Þ, provided Remark 2:5: In theorem above yð f Þ vanishes as an element of Hmk that M is compact or is an ANR.

3. Key Lemma and Proof of Theorem In this section we denote by M k a compact, connected generalized manifold of dimension k 4 n, by T: X ! X the fixed point free involution defined by Tðx; y; zÞ ¼ ðx; z; yÞ, where X ¼ S n  M k  M k . Letting D ¼ DðM k Þ be the ? Every open subset of a generalized manifold is again a generalized manifold.

104

CARLOS BIASI ET AL.

diagonal in M k  M k , we have in X two generalized submanifolds: S n  ð y0 ; y0 Þ and S n  DðM k Þ; which are both invariant under T and whose orbit spaces: S n  ð y0 ; y0 Þ=T ¼ P n  ð y0 ; y0 Þ and

S n  DðM k Þ=T ¼ P n  DðM k Þ

are generalized submanifolds in X=T of dimension n and n þ k, respectively. The following result is a particular generalization of [1, Lemma 32.3] and will be fundamental in the proof of the Theorem 1.1. LEMMA 3.1. Let j: P n  DðM k Þ,!X=T be the inclusion and let jk 2 H k ðX=T Þ be the Poincare´ dual of j ½P n  DðM k Þ . Then the kth Stiefel–Whitney class of the stable normal bundle of j is given by wk ð jÞ ¼ j ðjk Þ ¼

k X

ckm  wm ðM Þ;

m¼0

where c is the nonzero element of H1 ðP n Þ. Proof. The equality wk ð jÞ ¼ j ðjk Þ follows immediately from Theorem 2.4, since P n  DðM k Þ is compact and j: P n  DðM k Þ ! X=T is a proper topological embedding. Consider now the following commutative diagram

where i1 and i2 are inclusions. By using that j ðwðX=T ÞÞ ¼ j1 ðwðX=T ÞÞ  j2 ðwðX=T ÞÞ, we have wð jÞ ¼ j ðwðX=T ÞÞ ^ w ðP n  DÞ ¼ j1 ðwðX=T ÞÞ  j2 ðwðX=T ÞÞ ^ w ðP n Þ  w ðDÞ ¼ j1 ðwðX=T ÞÞ ^ w ðP n Þ  j2 ðwðX=T ÞÞ ^ w ðDÞ ¼ wð j1 Þ  wð j2 Þ;

ð3:1Þ

i.e., wð jÞ ¼ wð j1 Þ  wð j2 Þ and then it suffices to compute wð j1 Þ and wð j2 Þ. In fact, for wð j1 Þ, we observe that P n is a differentiable manifold, then we have wð j1 Þ ¼ wðnÞ, where n is the normal bundle to P n  ð y0  y0 Þ in X=T and follows that n X wð j1 Þ ¼ cm  1 m¼0

¼ 1  1 þ c  1 þ    þ ck  1 þ    þ c n  1:

ð3:2Þ

A BORSUK–ULAM THEOREM FOR MAPS

105

For wð j2 Þ, let U be an open neighborhood of x0 not containing any antipodal pair of points and denotes V ¼ U  M k  M k the open neighborhood in X. Then p: V ! V=T is a homeomorphism and follows from Theorem 2.3 that wð j2 Þ ¼ wð j2V=T Þ and wðiV Þ ¼ wðiÞ, where j2V=T ¼ j2 : ½x0  D ! V=Tð X=T Þ and iV ¼ i: ½x0  D ! Vð XÞ. Let us consider the following commutative diagram

Since p is a homeomorphism, from Remark 2.1 we have p wðV=T Þ ¼ wðV Þ, then wð j2 Þ ¼ wð j2V=T Þ ¼ ð j2V=T Þ ðwðV=T ÞÞ ^ w ðx0  DÞ ¼ ðpiV Þ ðwðV=T ÞÞ ^ w ðx0  DÞ ¼ ðiV Þ p ðwðV=T ÞÞ ^ w ðx0  DÞ ¼ ðiV Þ wðV Þ ^ w ðx0  DÞ ¼ wðiV Þ ¼ wðiÞ; i.e., wð j2 Þ ¼ wðiÞ. Moreover, since DðM k Þ ffi M k , by using the definition of the stable normal bundle of i, we have wðiÞ ¼ 1  wðM Þ and then wð j2 Þ ¼ 1  wðM Þ:

ð3:3Þ

Follows from (3.1), (3.2) and (3.3) that wk ð jÞ ¼

k X

wkm ð j1 Þ  wm ð j2 Þ

m¼0

¼

k X ðckm  1Þ  ð1  wm ðM ÞÞ m¼0

¼ ck  1 þ ck1  w1 ðM Þ þ    þ 1  wk ðM Þ ¼

k X

ckm  wm ðM Þ;

m¼0

This completes the proof.

&

Now let s: S n ! X be the map given by sðxÞ ¼ ðx; f ðxÞ; f ðxÞÞ. Then s induces a map s: P n ! X=T defined by sð½x Þ ¼ ½x; f ðxÞ; f ðxÞ . We denote by Bð f Þ the image of Að f Þ ¼ fx 2 S n ; f ðxÞ ¼ f ðxÞg under the natural map S n ! P n , then we have that Bð f Þ ¼ s1 ðP n  DðM k ÞÞ. In this conditions, we obtain the following lemma whose proof was presented by Munkholm in [2, Lemma 2.1], when M k is a compact topological manifold. Here, our purpose is emphasizing results also true for generalized manifolds.

106

CARLOS BIASI ET AL.

nk LEMMA 3.2. If s ðjk Þ 6¼ 0 then H ðBð f ÞÞ 6¼ 0. Proof. Let jk 2 H k ðX=T Þ be the Poincare´ dual of j ½P n  DðM k Þ . We first show for every neighborhood U of P n  DðM k Þ in X=T we have

jk 2 ImðH k ðX=T; X=T  U Þ ! H k ðX=T ÞÞ

ð3:4Þ

To prove this assertion we let V be an open neighborhood of P n  DðM k Þ such that V  U. Thus (3.4) follows from the commutative diagram

 ech duality (also true for generalized manifolds) in where g U denotes the Alexander–C the sense of [13, (6.2.17)], i is the natural transformation from H to H (see [13, p. 289]) and all the unlabelled maps are induced by appropriate inclusions. It  ech cohomology groups are naturally isomorphic is known that for ANR’s, the C to the singular cohomology groups (see [10, p. 230]). Next we prove: for every neighborhood V of Bð f Þ in P n we have ck 2 ImðH k ðP n ; P n  V Þ ! H k ðP n ÞÞ;

ð3:5Þ

where c is the generator of H1 ðP n Þ. Since for every neighborhood V of Bð f Þ in P n there is a neighborhood U of P n  DðM k Þ in X=T, with s1 ðU Þ  V, it is sufficient to prove (3.5) with V ¼ s1 ðU Þ, where U is a neighborhood of P n  DðM k Þ in X=T; and in this case the assertion follows immediately from the commutative diagram

using (3.4) and the hypothesis that s ðjk Þ ¼ ck . nk Now, assume that H ðBð f ÞÞ ¼ 0; then c nk maps to zero under the composition nk nk 1 H nk ðP n Þ ffi i H ðP n Þ ! H ðBð f ÞÞ;

therefore, by the definition of H there is an open neighborhood U of Bð f Þ in P n such that c nk maps to zero under H nk ðP n Þ ! H nk ðU Þ, i.e., we have c nk 2 kerðH nk ðP n Þ ! H nk ðU ÞÞ ¼ ImðH nk ðP n ; U Þ ! H nk ðP n ÞÞ: Using (3.5) and (3.6), with V closed and V  U we get that c n ¼ ck ^ c nk 2 ImðH n ðP n ; U [ ðP n  V ÞÞ ! H n ðP n ÞÞ;

ð3:6Þ

A BORSUK–ULAM THEOREM FOR MAPS

107

since H n ðP n ; U [ ðP n  V ÞÞ ¼ H n ðP n ; P n Þ ¼ 0 this gives the desired contradiction and Lemma 3.2 is proved. & Remark 3:3: The Lemma 3.2 reduces the proof of Theorem 1.1 to a consideration nk of s ðjk Þ, since H ðBð f ÞÞ 6¼ 0 implies that dim Að f Þ 5 n  k. The following result; which verifies under which conditions s ðjk Þ is not zero; has already been proved by Conner and Floyd in [1], in the case that M k is a differentiable manifold. The proof that we presented has been adapted for generalized manifolds. It is important to emphasize that here Lemma 3.1 is essential, as Lemma 32.3 has been in [1]. LEMMA 3.4. ðiÞ If n > k, then s ðjk Þ 6¼ 0 ðiiÞ If n ¼ k and f : H n ðM n Þ ! H n ðS n Þ is trivial then s ðjk Þ 6¼ 0: Proof. We observe that s ðjk Þ depends only on the homotopy class of f: S n ! M k . Hence, we may as well take f constant on the southern hemisphere; that is, f ðEn Þ ¼ y0 2 M k . Consider S n1  S n as the equator; thus f ðS n1 Þ ¼ y0 . We thus have the commutative diagram

where s1 ð½x Þ ¼ ½ðx; y0 ; y0 Þ for x 2 P n1 . Then, using the above diagram and the Lemma 3.1, we have j1 s ðjk Þ ¼ s1 j ðjk Þ ¼ s1 ðck  1 þ    þ 1  wk ðM ÞÞ ¼ j1 ðck Þ 2 H k ðP n1 Þ so, if n > k, j1 ðck Þ 6¼ 0 and consequently s ðjk Þ 6¼ 0. Consider now f: S n ! M n with f : H n ðM n Þ ! H n ðS n Þ trivial. We may continue to require f: ðS n ; En Þ ! ðM n ; y0 Þ. Now consider the equivariant? map F: S n ! M n  M n given by FðxÞ ¼ ð f ðxÞ; f ðxÞÞ. Then FðS n Þ  M n _ M n ¼ ðM n  y0 Þ [ ð y0  M n Þ, since either x or x is always in En . Moreover, the involution t: M n  M n ! M n  M n given by tð y; zÞ ¼ ðz; yÞ is such that tðM n _ M n Þ  M n _ M n . Then, we may consider the equivariant map F: ðS n ; AÞ ! ðM n _ M n ; tÞ, where A: S n ! S n is the antipodal map. ?

Let X and Y are topological spaces with involutions T: X ! X and S: Y ! Y. A map f from X to Y is equivariant if Sf ¼ f T.

108

CARLOS BIASI ET AL.

If F: P n ! M n _ M n =t ¼ M n is the map between orbit spaces induced by F; then F : H n ðM n Þ ! H n ðP n Þ is trivial:

ð3:7Þ

In fact, the assertion follows immediately from the commutative diagram

since f : H n ðM n Þ ! H n ðS n Þ is trivial, by hypothesis. Consider the composition i1

i2

P n  ð y0 ; y0 Þ ! S n  ðM n _ M n Þ=T ! S n  M n  M n =T ¼ X=T Then, we have i1 i2 ðjn Þ ¼ c n  1:

ð3:8Þ

We see this via the commutative diagram

and note that from Lemma (3.1) j ðjn Þ ¼ c n  1 þ    þ 1  wn ðM Þ 2 H n ðP n  DÞ. Denotes Y ¼ S n  ðM n _ M n Þ and let s1 : P n ! Y=T be given by s1 ð½x Þ ¼ ½ðx; f ðxÞ; f ðxÞÞ . Then s: P n ! X=T is given by composition s ¼ i2 s1 . Thus, we have only to show that s1 ði2 ðjn ÞÞ 6¼ 0. We have that s1 : H n ðY=T Þ ! H n ðP n Þ is an epimorphism, then there is gn 2 H n ðY=T Þ such that s1 ðgn Þ ¼ c n . Since i1 ðgn Þ ¼ c n  1, where i1 : H n ðY=T Þ ! H n ðP n ; ð y0 ; y0 ÞÞ, follows from (3.8) that i1 ðgn Þ ¼ i1 i2 ðjn Þ and in this case i1 ðgn þ i2 ðjn ÞÞ ¼ 0, i.e., gn þ i2 ðjn Þ 2 kerði1 Þ ¼ Imðk Þ;

ð3:9Þ

A BORSUK–ULAM THEOREM FOR MAPS

109

where k : H n ðY=T; P n  ð y0 ; y0 ÞÞ ! H n ðY=T Þ is induced by inclusion k: Y=T ! ðY=T; P n  ð y0 ; y0 ÞÞ. Now, we show in the diagram

that s1 k ¼ 0, where b: S n  ðM n _ M n Þ=T ¼ Y=T ! M n _ M n =t ¼ M n is induced by projection S n  ðM n _ M n Þ ! M n _ M n . Let us consider next the commutative diagram

where aðx; yÞ ¼ ½x; y; y0 , a0 ð yÞ ¼ ½y; y0 , and b0 is projection. It is easy to see that a is a relative homeomorphism. Since M n is ENR, follows from [13, 6.6.5] that a : H n ðY=T; P n  ð y0 ; y0 ÞÞ ! H n ðS n  M n ; S n  y0 Þ is an isomorphism. ða0 Þ and ðb0 Þ are easily seen to be isomorphisms: and we have that b : H n ðM n _ M n =t; ð y0  y0 ÞÞ ! H n ðY=T; P n  ð y0 ; y0 ÞÞ is an isomorphism. Since F ¼ 0 by (3.7), it follows from (3.10) that s1 k ¼ 0. Thus, from (3.9) we get s1 ðgn þ i2 ðjn ÞÞ ¼ 0, therefore s ðjn Þ ¼ s1 i2 ðjn Þ ¼ s1 ðgn Þ ¼ c n 6¼ 0: This completes the proof.

&

Proof of Theorem 1.1. If M k is a compact and connected generalized manifold, using Lemma 3.4 and Lemma 3.2 we have that H nk ðBð f ÞÞ 6¼ 0; since dim Að f Þ 5 dimðBð f ÞÞ, we conclude that dim Að f Þ 5 n  k, for n 5 k. & Remark 3:5: If M k is an open generalized manifold, since wðM Þ and wð f Þ have been defined for any generalized manifolds (Remark 2.2), using singular homology and cohomology with closed support, it is possible to prove Lemmas 3.1, 3.2 and 3.4 an therefore the theorem holds for all generalized manifolds.

Acknowledgement The second author was supported by FAPESP. The third author was supported by CAPES.

110

CARLOS BIASI ET AL.

References 1. Conner, P. E. and Floyd, E. E.: Differentiable Periodic Maps, Springer-Verlag, Berlin, 1964. 2. Munkholm, H. J.: A Borsuk–Ulam theorem for maps from a sphere to compact topological manifold, Illinois J. Math. 13 (1969), 116–124. 3. Biasi, C., Daccach, J., and Saeki, O.: A primary obstruction to topological embeddings for maps between generalized manifolds, Pacific J. Math. 197(2) (2001), 275–289. 4. Borel, A.: The Poincare´ duality in generalized manifolds, Michigan Math. J. 4 (1957), 227–239. 5. Borel, A. et al.: Seminar on Transformation Groups, Ann. of Math. Stud. 46, Princeton Univ. Press, Princeton, N.J., 1960. 6. Bredon, G. E.: Wilder manifolds are locally orientable, Proc. Nat. Acad. Sci. U.S.A 63 (1969), 1079–1081. 7. Bredon, G. E.: Sheaf Theory, Grad. Texts in Math. 170, Springer-Verlag, New York, 1997. 8. Bryant, J., Ferry, S., Mio, W., and Weinberger, S.: Topology of homology manifolds, Bull. Amer. Math. Soc. 28 (1993), 324–328. 9. Bryant, J., Ferry, S., Mio, W., and Weinberger, S.: Topology of homology manifolds, Ann. of Math. 143 (1996), 435–467. 10. Greenberg, M. J.: Lectures on Algebraic Topology, W. A. Benjamin, New York, Amsterdam, 1967. 11. Hurewicz, W. and Wallman, H.: Dimension Theory, Princeton Math. Ser. 4, Princeton Univ. Press, Princeton, 1948. 12. Milnor, J. and Stasheff, J.: Characteristic Classes, Ann. of Math. Stud. 76, Princeton Univ. Press, Princeton, N.J., 1974. 13. Spanier, E. H.: Algebraic Topology, McGraw-Hill, New York, 1966. 14. Thom, R.: Espaces fibre´s em sphpe`re et carre´s de Steenrod, Ann. Sci. Ecole Norm. Sup. 69 (1952), 109–181. 15. Wilder, R. L.: Topology of Manifolds, Amer. Math. Soc. Colloq. Publ. 32, Amer. Math. Soc., New York, 1949.