aa r X i v : . [ m a t h . C O ] F e b Borsuk-Ulam type theorems for manifolds
Oleg R. Musin ∗ Abstract
This paper establishes a Borsuk-Ulam type theorem for PL-manifoldswith a finite group action, depending on the free equivariant cobordismclass of a manifold. In particular, necessary and sufficient conditions areconsidered for a manifold with a free involution to be a Borsuk-Ulam type.
Keywords:
Borsuk - Ulam theorem, group action, equivariant cobordism.
The Borsuk-Ulam theorem states that any continuous antipodal f : S n → R n has zeros. (A map f is called antipodal if f ( − x ) = − f ( x ).) One of the mostinteresting proofs of this theorem is B´ar´any’s geometric proof [1] (see also [9,Sec. 2.2]). A very similar proof was given in [10], and [13] has several morereferences for proofs of this type.Let X = S n × [0 , , X = S n × { } , and X = S n × { } . Let τ ( x, t ) = ( − x, t ),where ( x, t ) ∈ X , x ∈ S n , and t ∈ [0 , τ is a free involution on X .The first step of B´ar´any’s proof is to show that any continuous antipodal(i.e. F ( τ ( x )) = − F ( x )) map F : X → R n can be approximated by “sufficientlygeneric” antipodal maps (see [9, Sec. 2.2]).Let f i : S n → R n , where i = 0 ,
1, be antipodal generic maps. Let F ( x, t ) = tf ( x ) + (1 − t ) f ( x ). Since F is generic, the set Z F := F − (0) is a manifold ofdimension one. Then Z F consists of arcs { γ k } with ends in Z f i := Z F T X i = f − i (0) and cycles which do not intersect X i . Note that τ ( Z F ) = Z F and τ ( γ i ) = γ j with i = j . Therefore, ( Z F , Z f , Z f ) is a Z - cobordism . It is nothard to see that Z f is Z -cobordant to Z f if and only if | Z f | = | Z f | (mod 4) . To complete the proof, take f as the standard orthogonal projection of S n onto R n : f ( x , . . . , x n , x n +1 ) = ( x , . . . , x n ) , where x + . . . + x n +1 = 1 . Since | Z f | = 2, we have | Z f | = 2 (mod 4) . This equality shows that for anyantipodal generic f the set Z f = f − (0) is not empty.Our analysis of this proof shows that it can be extended for a wide class ofmanifolds. For instance, consider two-dimensional orientable manifolds N = M g ∗ Research supported in part by NSF grant DMS-0807640 and NSA grant MSPF-08G-201.
1f even genus g and non-orientable manifolds N = P m with even m . Withoutloss of generality, we can assume that N is “centrally symmetric” embedded to R k , where k = 3 for N = M g and k = 4 for N = P m . That means A ( N ) = N ,where A ( x ) = − x for x ∈ R k . Then T := A | N : N → N is a free involution.It can be shown that there is a projection of N ⊂ R k into a 2-plane R passingthrough the origin 0 with | Z f | = 2 . (See details in Corollary 1.)Actually, B´ar´any’s proof can be (almost word for word) applied for N .Namely, the following statement holds: Let N = M g with even g or N = P m with even m . Then for any continuous h : N → R there is x ∈ N such that h ( T ( x )) = h ( x ) . Note that N can be represented as a connected sum M M . This statementcan be extended for N = M M with any closed manifold M (see Corollary 1). We see that one of the most important steps here is the existence of a genericequivariant f : N n → R n with | Z f | = 2 (mod 4) . This approach works for any free group action. Namely, for both technicalsteps a “generic approximation lemma” and a “( Z F , Z f , Z f ) cobordism lemma”can be extended for free actions of a finite group G (Section 2).Let G be a finite group acting free on a closed connected PL-manifold M m and linearly on R n . In Section 3 we show that a Borsuk-Ulam type theoremfor M , depending on the free equivariant cobordism class of M (Theorem 1).For the case m = n , Theorem 1 shows that if there is a continuous equivarianttransversal to zeros h : M n → R n with | Z h | = | G | (mod 2 | G | ), then for anycontinuous equivariant f : M n → R n the zero set Z f is not empty.In Section 4, this approach is applied to the classical case: manifolds with freeinvolutions. There we give necessary and sufficient conditions for Z -manifoldsto have a Borsuk-Ulam type theorem (Theorem 2).Lemma 4 shows that the zero set Z f is invariant up to G -cobordisms. Basedon this fact, in Section 5 we define homomorphisms µ of free G -cobordism groupswhich can be considered as obstructions for G -maps. Theorem 3 shows that fora free G -manifold M m and a given linear action G on R n , if this invariant isnot zero in the free equivariant cobordisms, then for any continuous equivariant f : M m → R n the set Z f is not empty. In this section also we also provide someapplications of Theorem 3 for the case G = ( Z ) k .The main goal of this paper is to show that the geometric proof gives amethod for checking whether a G -manifold M is of the Borsuk-Ulam type.Namely, from Theorem 1 and its extension Corollary 2, it follows that if thereexists a generic equivariant h : M m → R n with [ Z h ] = 0 in the correspondentgroup of cobordisms, then for any continuous equivariant f : M m → R n the setof zeros Z f = ∅ . It is not hard to prove the corollary using the standard techniques developed in [4].However, we couldn’t find this kind statements in the Borsuk-Ulam theorem literature. Generic G -maps If G is a group and X is a set, then a group action of G on X is a binary function G × X → X denoted ( g, x ) → g ( x ) which satisfies the following two axioms: (1)( gh )( x ) = g ( h ( x )) for all g, h in G and x in X ; (2) e ( x ) = x for every x in X (where e denotes the identity element of G ). The set X is called a G -set. Thegroup G is said to act on X .Let a finite group G act on a set X . We say that Y ⊂ X is a G -subset if g ( y ) ∈ Y for all g ∈ G and y ∈ Y , i.e. G ( Y ) = Y . Clearly, for any x ∈ X the orbit G ( x ) := { g ( x ) , g ∈ G } is a G -subset. Denote by Orb( X, G ) the set oforbits of G on X , i.e. Orb( X, G ) = { G ( x ) } Let G x := { g ∈ G | g ( x ) = x } denote the stabilizer (or isotropy subgroup ).Let X H = { x ∈ X | g ( x ) = x, ∀ g ∈ H } denote the fixed point set of a subgroup H ⊂ G . If H is isomorphic to G x = e , then X H = ∅ .Recall that a group action is called free if G x = { e } for all x , i.e. g ( x ) = x if only if g = e . Note that for a free action G on X each orbit G ( x ) consists of | G | points.Here we consider piece-wise linear (or PL , or simplicial ) G -manifolds X ,i.e. there is a triangulation Λ of X such that for any g ∈ G , any simplex σ of Λ is mapped bijectively onto the simplex g ( σ ). The triangulation Λ iscalled equivariant . Actually, any smooth G -manifold admits an equivarianttriangulation [8].Every free action of a finite group G on a compact PL-manifold X admitsan equivariant triangulation such that for each simplex there are no vertices inthe same orbit. Otherwise, we indeed can subdivide these simplices.A map f : X → Y of G -sets X and Y is called equivariant (or G -map ) if f ( g ( x )) = g ( f ( x )) for all g ∈ G, x ∈ X .For a PL G -space (in particular, for a PL G -manifold) we say that an equiv-ariant map f : X → R n is simplicial if f is a linear map for each simplex σ ∈ Λ.For an equivariant triangulation Λ, any simplicial map f : X → R n is uniquelydetermined by the set of vertices V (Λ). Indeed, for each simplex σ of Λ, f islinear, and therefore is determined by the vertices of σ .Any equivariant continuous map can be approximated by an equivariantsimplicial map. Lemma 1.
Let G be a finite group acting linearly on R n . Let X be a compactsimplicial G -space. Then for any equivariant continuous f : X → R n and ε > ,there is an equivariant simplicial map ¯ f : X → R n such that || f ( x ) − ¯ f ( x ) || ≤ ε for all x ∈ X .Proof. It is well known (see, for instance, [7, Section 2.2]) that any continu-ous map on a compact PL-complex can be approximated by simplicial maps.Therefore there exists a triangulation Λ and a simplicial map ˆ f such that || f ( x ) − ˆ f ( x ) || ≤ ε for all x ∈ X . Clearly, there exists an equivariant subdi-vision Λ ′ of Λ with G ( V (Λ)) ⊂ V (Λ ′ ). For v ∈ V (Λ ′ ), let ¯ f ( v ) := f ( v ). Then¯ f | V (Λ ′ ) defines a simplicial map with || f ( x ) − ¯ f ( x ) || ≤ ε .3et ρ : G → GL( n, R ) be a representation of a group G on R n . In otherwords, G is acting linearly on R n . Then F := ( R n ) G is a linear subspace of R n .Denote by L an invariant linear subspace of R n which is transversal to F . Then L G = { } . So without loss of generality we can consider only linear actions with( R n ) G = { } , i.e. linear actions on R n such that the fixed point set of theseactions is the origin.Let f : X → R n . Denote by Z f the set of zeros, i.e. Z f = { x ∈ X : f ( x ) = 0 } . In this paper we need generic maps with respect to Z f . Let X be a simplicial m -dimensional G -manifold. Set O ε := { v ∈ R n : || v || < ε } . A continuousequivariant f : X m → R n is called transversal to zeros if Z f is a manifold ofdimension m − n , and there is ε > v ∈ O ε , the sets f − ( v )and Z f are homeomorphic. Definition.
Given a finite group G . Let X m be a PL free G -manifold. LetΛ be an equivariant triangulation of X . We say that an equivariant simplicialmap f : X → R n is generic (with respect to zeros) if Z f does not intersect the( n − n −
1) of Λ. (Recall that the k -skeleton of Λ is the subcomplexΛ( k ) that consists of all simplices of dimension at most k .)All simplicial generic maps are transversal to zeros. Lemma 2.
Let G be a finite group acting linearly on R n with ( R n ) G = { } . Let X m be a PL compact m -dimensional free G -manifold with or without boundary.Let f : X → R n be an equivariant simplicial generic map. If n ≤ m , then Z f isan invariant submanifold of X of dimension m − n . Moreover, if the boundary ∂Z f is not empty, then it lies in ∂X .Proof. Note that Z f ⊂ X is a locally polyhedral surface consisting of ( m − n )-dimensional cells. This is because for each m -simplex σ we have the linear map f | σ : σ → R n , and Z f ∩ σ is defined by the generic linear equation f ( x ) = 0.Hence, the components of Z f are PL manifolds.For G -spaces, the Tietze-Gleason theorem states: Let a compact group G acton X with a closed invariant set A . Let G also acts linearly on R n . Then anyequivariant f : A → R n extends to f : X → R n (see [3, Theorem 2.3]).It is known from results of Bierstone [2] and Field [6] on G -transversalitytheory that the zero set is a stratified set. If G acts free on a compact smooth G -manifold X , then this theory implies that the set of non-generic equivariantsmooth f : X m → R n has measure zero in the space of all smooth G -maps. Letus extend this result as well as the Tietze-Gleason theorem for the PL case. Lemma 3.
Let G be a finite group acting linearly on R n with ( R n ) G = { } .Let X m be a PL compact free G -manifold with or without boundary. Let Λ bean equivariant triangulation of X . Let A be an invariant subset of V (Λ) or let A = ∅ . Let h : A → R n be a given equivariant generic map. Then the set ofnon-generic simplicial maps f : X → R n with f | A = h has measure zero in thespace of all possible equivariant maps f : V (Λ) → R n with f | A = h . roof. Let V ′ := V (Λ) \ A . Let C ∈ Orb( V ′ , G ) be an orbit of G on V ′ .Then | C | = p , where p := | G | . Since for any equivariant f : V (Λ) → R n and v ∈ V (Λ) we have f ( g ( v )) = g ( f ( v )), the vector u = f ( v ) ∈ R n yields the map f : C → R n .Suppose G has k orbits on V ′ . Take in each orbit C i ∈ Orb( V ′ , G ) a vertex x i . Denote by M G the space of all equivariant maps f : V ′ → R n . Then thespace M G is of dimension N = kn .Consider an ( n − σ ∈ Λ with vertices v , . . . , v n which are not allfrom A . Note that f ∈ M G is not generic on σ if 0 ∈ f ( σ ). In particular, thehyperplane in R n which is defined by vectors f ( v ) , . . . , f ( v n ) passes through theorigin 0. In other words, the determinant of n vectors f ( v ) , . . . , f ( v n ) equalszero.This constraint gives a proper algebraic subvariety s ( σ ) in M G . Let v , . . . , v d be vertices of σ which are not in A . Then s ( σ ) is of dimension N −
1. Indeed, u i = f ( v i ) , i = 1 , . . . , d , are vectors in R n . Then the equationdet( u , . . . , u d , h ( v d +1 ) , . . . , h ( v n )) = 0defines the subvariety s ( σ ) in M G of dimension N − s ( σ ) with dim σ = n − N −
1, it has measure zero in M G . From this it follows that the setof all non-generic f has measure zero in this space.Let M and M be closed m -dimensional simplicial manifolds with free ac-tions of a finite group G . Then an ( m +1)-dimensional simplicial free G -manifold W is called a free G -cobordism ( W, M , M ) if the boundary ∂W consists of M and M , and the action G on W respects actions on M i . Lemma 4.
Let G be a finite group acting linearly on R n with ( R n ) G = { } . Let ( W, M , M ) be a free G -cobordism. Let f i : M mi → R n , i = 0 , , be equivariantsimplicial generic maps. Then there is an equivariant simplicial generic F : W → R n with F | M i = f i , and such that ( Z F , Z f , Z f ) is a free G -cobordism.Proof. The existence of such a generic F follows from Lemma 3. Lemma 2implies that Z F is a G -submanifold of W . Clearly, g ( Z F ) = Z F and g ( Z f i ) = Z f i for all g ∈ G . Since ∂Z F = Z f F Z f is an ( m − n )-dimensional G -manifold, wehave a free G -cobordism ( Z F , Z f , Z f ). G -maps and a Borsuk-Ulam type theorem In this section we consider a Borsuk-Ulam type theorem for the case m = n . Definition.
Given a finite group G acting free on a closed PL-manifold M n and acting linearly on R n with ( R n ) G = { } . Let f : M n → R n be a continuousequivariant transversal to zeros map. Since Z f is a finite free G -invariant subsetof M , we have | Z f | = k | G | , where k = 0 , , , . . . Set deg G ( f ) := 1 if k is odd,and deg G ( f ) := 0 if k is even. 5 emma 5. Let G be a finite group acting linearly on R n with ( R n ) G = { } .Let ( W n +1 , M n , M n ) be a free G -cobordism. Let h i : M ni → R n , i = 0 , , becontinuous equivariant transversal to zeros maps. Then deg G ( h ) = deg G ( h ) . Proof.
Lemma 1 and Lemma 3 yield that for any h i : M ni → R n and ε > a i,ε such that || a i,ε ( x ) − h i ( x ) || ≤ ε for all x ∈ M i . If ε → Z i,ε := a − i,ε (0) → Z h i . This implies that for a sufficiently small ε there is anequivariant bijection between Z i,ε and Z h i . Therefore, deg G ( a i,ε ) = deg G ( h i ).Let ε be sufficiently small. Set f i := a i,ε . From Lemma 4 it follows thatthere is an equivariant simplicial generic F : W → R n with F | M i = f i such that( Z F , Z f , Z f ) is a free G -cobordism. We have m = n . Then Z F is a submanifoldof W of dimension one. Therefore Z F consists of arcs γ , . . . , γ ℓ with ends in Z f i and cycles which do not intersect M i .Since any continuous map s : γ k → γ k has a fixed point, G cannot act freeon γ k . Therefore, G acts free on the set of all arcs { γ k } . Moreover, the ends ofany arc cannot lie in the same orbit of G . From this it follows that | Z f | = | Z f | (mod 2 | G | ), i.e. deg G ( f ) = deg G ( f ). Theorem 1.
Let G be a finite group acting linearly on R n with ( R n ) G = { } .Let M n be a closed connected PL free G -manifold. If there is a closed PL free G -manifold N n which is free G -cobordant to M n and a continuous equivarianttransversal to zeros h : N n → R n with deg G ( h ) = 1 , then for any continuousequivariant f : M n → R n the zero set Z f is not empty.Proof. Let deg G ( h ) = 1. Suppose that Z f = ∅ . Since M is compact, there isan ε > || f ( x ) || ≥ ε for all x ∈ M . From Lemma 3 it follows thatthere exists a generic ˜ f such that || f ( x ) − ˜ f ( x ) || ≤ ε/ x ∈ M . Then || ˜ f ( x ) || ≥ ε/ x ∈ M and Z ˜ f = ∅ . Therefore, deg G ( ˜ f ) = 0. On the otherhand, Lemma 5 implies 0 = deg G ( ˜ f ) = deg G ( h ) = 1, a contradiction. Remark.
Actually, Lemma 5 immediately implies that the assumption in thetheorem:
There is a free G -manifold N n which is free G -cobordant to M n anda continuous equivariant transversal to zeros h : N n → R n with deg G ( h ) =1 is equivalent to the following statement: There is a continuous equivarianttransversal to zeros map h : M n → R n with deg G ( h ) = 1 . However, since theassumption in the theorem is more general, it sometimes can be more easilychecked for N which are free G -cobordant to M . (For instance, see our proofof Theorem 2.) In this section, we consider the classical case G = Z . Let M be a closedPL-manifold with a free simplicial involution T : M → M , i.e. T ( x ) = x and T ( x ) = x for all x ∈ M . For any Z -manifold ( M, T ) we say that a map f : M m → R n is antipodal (or equivariant) if f ( T ( x )) = − f ( x ).6 efinition. We say that a closed PL free Z -manifold ( M n , T ) is a BUT(Borsuk-Ulam Type) manifold if for any continuous g : M n → R n there is apoint x ∈ M such that g ( T ( x )) = g ( x ). Equivalently, if a continuous map f : M n → R n is antipodal, then Z f is not empty.Let us recall several facts about Z -cobordisms which are related to ourmain theorem in this section. We write N n for the group of unoriented cobor-dism classes of n -dimensional manifolds. Thom’s cobordism theorem says thatthe graded ring of cobordism classes N ∗ is Z [ x , x , x , x , x , x , . . . ] with onegenerator x k in each degree k not of the form 2 i −
1. Note that x k = [ RP k ].Let N ∗ ( Z ) denote the unoriented cobordism group of free involutions. Then N ∗ ( Z ) is a free N ∗ -module with basis [ S n , A ], n ≥
0, where [ S n , A ] is thecobordism class of the antipodal involution on the n -sphere [5, Theorem 23.2].Thus, each Z -manifold can be uniquely represented in N n ( Z ) in the form:[ M, T ] = n X k =0 [ V k ][ S n − k , A ] . Theorem 2.
Let M n be a closed connected PL-manifold with a free simplicialinvolution T . Then the following statements are equivalent:(a) M is a BUT manifold.(b) M admits an antipodal continuous transversal to zeros map h : M n → R n with deg Z ( h ) = 1 .(c) [ M n , T ] = [ S n , A ] + [ V ][ S n − , A ] + . . . + [ V n ][ S , A ] in N n ( Z ) .Proof. (1) First we prove that (c) is equivalent to (b).
Let[
M, T ] = [ V ][ S n , A ] + [ V ][ S n − , A ] + . . . + [ V n ][ S , A ] . Denote by N := V × S n − ⊔ . . . ⊔ V n × S . Note that for any k , where 0 ≤ k < n , the standard antipodal embeddingof S k into R n has no zeros. This implies that there is a generic antipodal p : N n → R n with Z p = ∅ . If [ V ] = 0, then M is free Z -cobordant to N .Therefore, Lemma 5 yields that for any generic antipodal f : M → R n we havedeg Z ( f ) = deg Z ( p ) = 0. This contradicts (b), and thus [ V ] = 1 ∈ Z .On the other hand, if [ V ] = 1, then M is free Z -cobordant to L := S n ⊔ N .Take any generic antipodal h : M n → R n . Since for any generic antipodal g : L → R n we have deg Z ( g ) = 1, it again follows from Lemma 5 that deg Z ( h ) = 1. (2) Theorem 1 yields: (b) implies (a). (3) (a) implies (c).
In fact, it follows from [12, Theorem 3]. This theorem yieldsthat if w n ( M/ Z ) = 0 ∈ H n ( M/ Z , Z ), then there is an antipodal continuousmap h : M n → S n − , i.e. Z h = ∅ . Therefore, w n ( M/ Z ) = 0. It holds if andonly if we have (c). I would to thank Alexey Volovikov who noted this theorem. He as well as Pavle Blagojevi´cand Roman Karasev also sent me another proof of [12, Theorem 3]. emark. Actually, the list of equivalent versions in Theorem 2 can be contin-ued. Tucker’s lemma is a discrete version of the Borsuk-Ulam theorem:
Let Λ be any equivariant triangulation of S n . Let L : V (Λ) → { +1 , − , +2 , − , . . . , + n, − n } be an equivariant (or Tucker) labelling, i.e. L ( T ( v )) = − L ( v ) ). Then there ex-ists a complementary edge in Λ such that its two vertices are labelled by oppositenumbers (see [9, Theorem 2.3.1]).Let M be as in Theorem 2. Using the same arguments as in [9, Theorem2.3.2], it can be proved that the following statement is equivalent to (a): (d) For any equivariant labelling of an equivariant triangulation of M there isa complementary edge. In fact, any equivariant labelling L defines a simplicial map f L : M → R n .Indeed, let e , . . . , e n be an orthonormal basis of R n . We define f L : Λ → R n for v ∈ V (Λ) by f L ( v ) = e i if L ( v ) = i and f L ( v ) = − e i if L ( v ) = − i . In thepaper [11], it is shown that M is a BUT manifold if and only if (e) There exist an equivariant triangulation Λ of M and an equivariant labellingof V (Λ) such that f L : Λ → R n is transversal to zeros and the number ofcomplementary edges is k + 2 , where k = 0 , , , . . . Lyusternik and Shnirelman proved in 1930 that for any cover F , . . . , F n +1 of the sphere S n by n + 1 closed sets, there is at least one set containing apair of antipodal points (that is, F i ∩ ( − F i ) = ∅ ). Equivalently, for any cover U , . . . , U n +1 of S n by n + 1 open sets, there is at least one set containing a pairof antipodal points [9, Theorem 2.1.1]. By the same arguments it can be shownthat M is a BUT manifold if and only if (f ) M is a Lusternik-Shnirelman type manifold, i.e. for any cover F , . . . , F n +1 of M n by n + 1 closed (respectively, by n + 1 open) sets, there is at least one setcontaining a pair ( x, T ( x )) . Denote by cat( X ) the Lusternik-Shnirelman category of a space X , i.e. thesmallest m such that there exists an open covering U , . . . , U m +1 of X with each U i contractible to a point in X . It is not hard to prove the last statement inthis Remark: (g) M is a BUT manifold if and only if cat( M/T ) = cat( RP n ) = n . Usually, it is not easy to find cat( X ). Here, Theorems 2-4 yield interestingpossibilities to find lower bounds for cat( M/G ). Corollary 1.
Let M be any closed manifold. Then for the connected sum M M there exists a “centrally symmetric” free involution such that M M isa BUT manifold. roof. The Whitney embedding theorem states that any smooth or simplicial n -dimensional manifold can be embedded in Euclidean 2 n -space. Consider anembedding of M in R n with coordinates ( x , . . . , x n ). Let M − (respectively, M + ) denote the set of points in M ⊂ R n with x < x > M be the set of points in M with x = 0. Without loss of generality we canassume that M is embedded in R n in such a way that M − is homeomorphicto an open n -ball and M is a sphere x + . . . + x n +1 = 1 with x n +2 = . . . = x n = 0. Then the central symmetry s ( s ( x ) = − x ) is well defined on X := s ( M + ) S M S M + as a free involution, and X is homeomorphic to M M .Consider the projection h of X onto the n -plane x n +1 = . . . = x n = 0, i.e. h ( x , . . . , x n ) = ( x , . . . , x n ). Since Z h ⊂ M , we have Z h = Z t , where t := h | M . On the other hand, t : M → R n is an orthogonal projection of S n . Thus, | Z h | = 2 and Theorem 2(b) implies that M M is a BUT manifold. G -maps in cobordisms In the previous sections, we considered equivariant maps f : M m → R n with m = n . Now we extend this approach to the case m ≥ n .Consider closed PL manifolds with an H -structure (such as an orientation).One can define a “cobordism with H -structure”, but there are various technical-ities. In each particular case, cobordism is an equivalence relation on manifolds.A basic question is to determine the equivalence classes for this relationship,called the cobordism classes of manifolds. These form a graded ring called thecobordism ring Ω H ∗ , with grading by dimension, addition by disjoint union, andmultiplication by cartesian product.Let Ω H ∗ ( G ) denote the PL cobordism group with H -structure of free simpli-cial actions of a finite group G . Let ρ : G → GL( n, R ) be a representation of agroup G on R n which also has H -structure. Lemma 4 shows that for a genericsimplicial equivariant map f : M m → R n the cobordism class of the manifold Z f is uniquely defined up to cobordisms and so well defines a homomorphism µ Gρ : Ω Hm ( G ) → Ω Hm − n ( G ) . Remark.
Note that the homomorphism µ Gρ : Ω Hm ( G ) → Ω Hm − n ( G ) depends onlyon a representation of a group G on R n . For some groups, this homomorphismis known in algebraic topology. For instance, if G = Z and H = O (unorientedcobordisms, i.e. Ω O ∗ ( Z ) = N ∗ ( Z )), then µ = ∆ k , where ∆ is called the Smithhomomorphism . Conner and Floyd [5, Theorem 26.1] give the following defini-tion of ∆:
Let T be a free involution on a closed manifold M . For n ≥ m thereexists an antipodal generic map f : M m → S n . Let X m − = f − ( S n − ) . Thenthe map ∆ : Ω Om ( Z ) → Ω Om − ( Z ) which is defined by [ M m , T ] → [ X m − , T | X ] is a homomorphism and ∆ does not depend on n and f . The invariant µ Gρ is an obstruction for the existence of equivariant maps f : M → R n \ { } . Namely, we have the following theorem.9 heorem 3. Let M m be a closed PL G -manifold with a free action τ . Let ρ bea linear action of G on R n . Let us assume that actions, manifolds, and mapsare with H -structure. Suppose that µ Gρ ([ M, τ ]) = 0 in Ω Hm − n ( G ) . Then for anycontinuous equivariant map f : M m → R n the set of zeros Z f is not empty.Proof. Actually the proof of the theorem and of the corollary are almost wordfor word the same as the proof of Theorem 1. Let us suppose that f − (0) = ∅ .Since M is compact, there is an ε > || f ( x ) || ≥ ε for all x ∈ M .From Lemma 1 and Lemma 3, it follows that there exists a generic h such that || f ( x ) − h ( x ) || ≤ ε/ x ∈ M . By Lemma 4 we have µ Gρ ([ M, τ ]) = [ Z h ] G = 0in Ω Hm − n ( G ). Then Z h is a submanifold of M of dimension m − n .Since Z h = ∅ , there is x ∈ M such that h ( x ) = 0. Thus, ε/ ≥ || f ( x ) − h ( x ) || = || f ( x ) || ≥ ε >
0, a contradiction.
Corollary 2.
Let manifolds, actions, and maps be as above. Suppose that thereis a continuous equivariant transversal (in zeros) map h : M m → R n such thatthe set Z h is a closed submanifold in M of codimension n and [ Z h ] G = 0 in Ω Hm − n ( G ) . Then for any continuous equivariant map f : M → R n the set ofzeros Z f is not empty.Proof. By Lemma 3, for any ε > a ε such that || a ε ( x ) − h ( x ) || ≤ ε for all x ∈ M . If ε →
0, then Z ε := a − ε (0) → Z . This impliesthat for a sufficiently small ε , there is a homeomorphism between Z ε and Z h .Therefore, µ Gρ ([ M, τ ]) = [ Z h ] G = 0 in Ω Hm − n ( G ) . Let G = Z and ρ ( x ) = − x . Denote by µ n := µ Z ρ . Then we have µ n : N m ( Z ) → N m − n ( Z ) . If u ∈ N m ( Z ) is given explicitly, then we can easily find µ n ( u ). Lemma 6. µ n ([ M m , T ]) = µ n m X k =0 [ V k ] [ S m − k , A ] ! = m − n X k =0 [ V k ] [ S m − n − k , A ] . Proof.
Since Z f i = F − (0) T M i in Lemma 4 are cobordant to each other, wecan consider the standard projection P r : S d → R n . Then P r − (0) = S d − n .This yields µ n (cid:0) [ S d , A ] (cid:1) = [ S d − n , A ]. It is clear that the lemma follows from thisequality.Lemma 6 and Theorem 3 imply the following corollary. Corollary 3.
Let [ M m , T ] = [ V m − n ][ S n , A ]+[ V m − n +1 ][ S n − , A ]+ . . . +[ V m ][ S , A ] in N m ( Z ) with [ V m − n ] = 0 in N m − n . Then for any antipodal continuous map f : M m → R n the set Z f is not empty. G = ( Z ) k = Z × . . . × Z . Notethat N ∗ (( Z ) k ) = N ∗ ( Z ) ⊗ . . . ⊗ N ∗ ( Z ) [5, Sec. 29]. In other words, N ∗ (( Z ) k )is a free N ∗ -module with generators { γ i ⊗ . . . ⊗ γ i k } , i , . . . , i k = 0 , , . . . , where γ i := [ S i , A ] ∈ N i ( Z ). Corollary 4.
Let M be a closed PL manifold with a free action Ψ of ( Z ) k .Let [ M m , Ψ] = P [ V i ,...,i k ] γ i ⊗ . . . ⊗ γ i k in N m (( Z ) k ) with [ V i ,...,i k ] = 0 in N ∗ . Consider ( Z ) k with generators T , . . . , T k acting on R i ⊕ . . . ⊕ R i k by T ℓ ( x ) = − x for x ∈ R i ℓ . Then for any continuous equivariant map f : M m → R i ⊕ . . . ⊕ R i k the set Z f is not empty. Let ˜ G (4 ,
2) be the oriented Grassmann manifold, that is, the space of alloriented 2-dimensional subspaces of R . Note that Z × Z acts on ˜ G (4 , x ∈ ˜ G (4 , T ( x ) changes the orientation of x and T ( x ) is theoriented orthogonal complement of an oriented 2-plane x in R .It is well known that ˜ G (4 ,
2) is diffeomorphic to S × S [7, 3.2.3]. For( u, v ) ∈ S × S , we have T ( u, v ) = ( − u, − v ) and T ( u, v ) = ( u, − v ). Then forΨ = ( T , T ) we have: [ ˜ G (4 , , Ψ] = γ ⊗ γ in N ( Z × Z ). Then Corollary 4(or Theorem 1) yields: Corollary 5.
Let Z × Z act on R = R ⊕ R by T ( u, v ) = ( − u, − v ) and T ( u, v ) = ( u, − v ) . Then for any equivariant continuous f : ˜ G (4 , → R theset Z f is not empty. Acknowledgment.
I wish to thank Arseniy Akopyan, Imre B´ar´any, PavleBlagojevi´c, Mike Field, Jiˇr´ı Matouˇsek, Alexey Volovikov, and G¨unter Zieglerfor helpful discussions and comments. I am most grateful to Roman Karasevand to the anonymous referee for several critical comments and corrections.
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