COINCIDENCE THEOREMS FOR MAPS OF FREE Z p -SPACES

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1 COINCIDENCE THEOREMS FOR MAPS OF FREE Z p -SPACES EDIVALDO L DOS SANTOS AND FRANCIELLE R DE C COELHO 2 Abstract Let us consider Z p, p a prime number, acting freely on Hausdorff paracompact topological space X and let Y be a k-dimensional metrizable space (or k-dimensional CW-complex) In this paper, by using the genus of X; gen (X, Z p ), we prove a Z p -coincidence theorem for maps f : X Y Such theorem generalizes the main theorem proved by Aarts, Fokkink and Vermeer in [] Key words: Z p -coincidence point, free Z p -action, genus of Z p -space Introduction The classic Borsuk-Ulam theorem says that every map of S n into the euclidean k-dimensional space R k has an antipodal coincidence if n k This result can be generalized in many ways: S n and R k can be replaced by more general spaces X and Y, and the antipodal action Z 2 on S n can be replaced by actions of others groups In one of these generalizations Aarts, Fokkink and Vermeer [, Theorem ] proved that if i : X X is a fixed-point free involution of a normal space X with color number n+2 and k is a natural number then for every k-dimensional cone CW-complex Y and every continuous map ϕ : X Y there is an Z 2 -coincidence, whenever n 2k; and this result is the best possible Let us observe that for X = S n the result was obtained independently by Shchepin in [8] In this paper, requiring that X is a Hausdorff paracompact space, we generalized the Aarts, Fokkink and Vermeer s result for free Z p -actions, p prime Specifically, we prove 2000 Mathematics Subject Classification Primary 55M0, 55M20; Secondary 55M35 Universidade Federal de São Carlos, Departamento de Matemática, , São Carlos SP, Brazil, edivaldo@dmufscarbr The author was supported in part by CNPq of Brazil Grant number / and by FAPESP 2 Universidade Federal de Uberlândia, Faculdade de Matemática, , Uberlândia MG, Brazil, francielle@famatufubr The author was supported by CAPES

2 2 EDIVALDO L DOS SANTOS AND FRANCIELLE R DE C COELHO Theorem Let X be a Hausdorff paracompact space equipped with a free Z p -action generated by α : X X such that gen(x, Z p ) n + and let k be a natural number Then the following holds (a) If n > p k, then for every k-dimensional metrizable space Y and every continuous map f : X Y there is a Z p -coincidence point, ie, there is x X such that f(x) = f(α i (x)), i {, 2,, p } (b) If n = p k, then for every k-dimensional cone CW-complex Y and for every continuous map f : X Y there is a Z p -coincidence point (c) If n < p k and gen (X, Z p ) = n +, then there exists a k-dimensional cone CW-complex Y and a continuous map f : X Y such that f has no Z p -coincidence points In the case n = pk, we exhibit an interesting example showing that the result does not hold for the larger class of CW-complexes of dimension k Example 2 Consider Y = ps+p 2 s (the (s )-skeleton of the (ps+p 2)- simplex) and Y = Π p i= Y i, where Y i = Y, i and is the diagonal We have that Z p acts freely on Y and Y is Hausdorff, paracompact space Moreover, it follows from[2] and [2] that gen (Y, Z p ) = p(s ) + Define π : Y Y by π(y,, y p ) = y, (y,, y p ) Y and clearly π has no Z p -coincidence points From this, we conclude that the theorem does not hold in the case n = p k when Y is any CW -complex Remark 3 In the case that Y is a cone CW-complex, Theorem is the best possible Note that, if we consider X a Hausdorff paracompact free Z 2 - space with color number n + 2, by Theorem 23, we have that gen(x, Z 2 ) = n + and in this way, Theorem generalizes main result of [] Remark 4 Let us consider G = Z p and X satisfying the assumptions of [5, Theorem ] We have that H m+ (Z p, Z) 0, for all m odd, and since H i (X, Z) = 0, for 0 < i < m, by Proposition 29, gen (X, Z p ) m + Therefore, it follows from Theorem (i) that, whenever n > pk, for every continuous map f : X Y, with Y CW-complex k-dimensional, there is a Z p -coincidence point Then, in the case G = Z p and n > pk, Theorem includes the result proved by Gonçalves, Jaworowski, Pergher and Volovikov in [5] 2 Preliminaries Aarts, Brouwer, Fokkink and Vermeer, in [2], defined the genus, gen (X, G), in the sense of Švarc, as follows Let G be a finite group which acts freely on a space X Hausdorff paracompact Let G denote G\{e} We say that an open subset U of X is a

3 COINCIDENCE THEOREMS FOR MAPS OF FREE Z p-spaces 3 color if U g U = for all g G and we shall say that a cover U of X by colors is a coloring If (X, G) admits a finite coloring, then the color number col (X, G) is the minimal cardinality of a coloring If U is a color, then the set G U = g G g U is called a set of the first kind and G U is said to be generated by the color U As G is a group, the collection {g U g G} is pairwise disjoint The space X together with the group action is usually called a G-space Definition 2 Suppose that X is a G-space and let U be a color We say that a set G U is a set of the first kind The genus, gen (X, G), is defined as the minimal cardinality of a covering of X by sets of the first kind It follows from the definition that the genus in non-decreasing under equivariant maps Proposition 22 Let X and Y be free G-spaces Hausdorff paracompacts and let F : X Y be G-equivariant map Then, gen (X, G) gen (Y, G) Hartskamp [6] and Bogatyi [3, Theorem 5] proved independently the following result: Theorem 23 Suppose that X is a Hausdorff paracompact G-space The following statements are equivalent (i) gen (X, G) = n + ; (ii) col (X, G) = n + G Other papers in connection with Theorem 23 are the papers of Steinlein [0, ] Krasnosel skiǐ in [7], proved the following theorem: Theorem 24 gen (S n, Z p ) = n + For two simplicial spaces X and Y, recall that the join X Y is the simplicial space realized by all simplices [x, x 2,, x k, y, y 2,, y m ] for simplices [x, x 2,, x k ] and [y, y 2,, y m ] in X and Y, respectively If X and Y are G-spaces, then so is X Y The k-fold join G G G is the simplicial space realized by all [g, g 2,, g k ], with g i G, for i =,, k The G-action on the join is induced by g i gg i on the vertices Definition 25 The k-fold join G G G with the standard action is a G-space, denoted by S k G From [2], it follows the result: Theorem 26 Let X be a free G-space Hausdorff paracompact such that gen (X, G) k Then, there exists a G-equivariant map F : X S k G

4 4 EDIVALDO L DOS SANTOS AND FRANCIELLE R DE C COELHO In [9], Švarc obtained the following theorem: Theorem 27 Suppose that X is a Hausdorff paracompact G-space of dim X = n Then gen (X, G) n + In [2], Aarts, Brouwer, Fokkink and Vermeer proved that Theorem 28 Let G be a finite group (i) gen (SG k, G) = k (ii) Suppose that X is a free G-space paracompact Hausdorff (k 2)- connected Then, there exists a G-equivariant map F : SG k X and as a consequence, gen (X, G) k Volovikov, in [2], proved the following proposition: Proposition 29 Suppose that G = Z n p acts on X without fixed points If H i (X) = 0 for i N, then gen (X, G) N + 3 Proof of Theorem Proof (Case (a) n > pk) Let α : X X be a map that generates a free Z p -action on X with gen (X, Z p ) n + and let f : X Y be a continuous map Suppose, by contradiction, that for each x X, exists i, j {, 2,, p}, i j, satisfying f(α i (x)) f(α j (x)), where α p = id Consider Y = Π p i= Y i, where = {(y, y 2,, y p ) Π p i= Y i, y = y 2 = = y p } We have that Y is a metrizable space with dim Y p k Define σ : Y Y by σ(y, y 2,, y p ) = (y p, y,, y p ), (y, y 2,, y p ) Y, and φ : X Y by φ(x) = (f(α p (x)), f(α p 2 (x)),, f(α(x)), f(x)), x X Note that, σ generates a free Z p -action on Y and φ is a continuous map well-defined Moreover, φ is a Z p -equivariant map Since the genus is non-decreasing under equivariant maps, we have that gen (X, Z p ) gen (Y, Z p ) Now, since dim Y p k, it follows from Theorem 27 that (3) gen (Y, Z p ) p k + Therefore, gen (X, Z p ) gen (Y, Z p ) p k+ < n+, which contradicts gen (X, Z p ) n +

5 COINCIDENCE THEOREMS FOR MAPS OF FREE Z p-spaces 5 (Case (b) n = pk) The strategy used to show the case n = pk for a cone CW -complex Y, is the following: we shall show that the upper bound of equation (3) can be reduced by one, ie, we shall prove that gen (Y, Z p ) p k For this, let Y be a k-dimensional CW -complex, which is a cone CW - A [0, ] complex, ie, Y = CA =, where A is a CW -complex of dimension k and is the following equivalence relation: (a, ) (a, ), a, a A In this sense, we obtain coordinates for Y A point in Y is represented by class [a, u], with a A and u [0, ] We take Y = Π p i= Y i Define σ : Y Y by σ(y, y 2,, y p ) = (y p, y,, y p ), (y, y 2,, y p ) Y, or using coordinates, by σ([a, u ],, [a p, u p ]) = ([a p, u p ], [a, u ],, [a p, u p ]), a,, a p A e u,, u p [0, ] Note that, σ generates a free Z p -action on Y Lemma 3 gen (Y, Z p ) p k Proof Let Z = [0, ] [0, ] \ {(,, )} and let s : Z Z be given by s(u,, u p ) = (u p, u,, u p ), (u,, u p ) Z The projection π : Y Z defined by π([a, u ],, [a p, u p ]) = (u,, u p ), ([a, u ],, [a p, u p ]) Y, is well-defined, is continuous and s π = π σ Let us consider the following subsets of Z,

6 6 EDIVALDO L DOS SANTOS AND FRANCIELLE R DE C COELHO W = {2/3} [2/3, ] [2/3, ] [2/3, ] W 2 = [2/3, ] {2/3} [2/3, ] [2/3, ] W p = [2/3, ] [2/3, ] [2/3, ] {2/3} and we define W = {} [0, 2/3] [0, ] [0, ] [0, ] W 2 = {} [2/3, ] [0, 2/3] [0, ] [0, ] W p = {} [2/3, ] [2/3, ] [0, 2/3] W p = [0, 2/3] [0, ] [0, ] [0, ] {} W p 2 = [2/3, ] [0, 2/3] [0, ] [0, ] {} W p p = [2/3, ] [2/3, ] [2/3, ] [0, 2/3] {}, W = ( p i= W i) ( p j= W j ) ( p j= W p j ) We have that W is the union of p 2 = p + p (p ) closed subsets of Z ( Figure illustrates the cases p = 2 and p = 3): W 2/3 2/3 0 2/3 2/3 0 2/3 (a) case p = 2 (b) case p = 3 Figure

7 COINCIDENCE THEOREMS FOR MAPS OF FREE Z p-spaces 7 Define a retraction r : Z W as follows: In the right upper corner of Z, the retraction r is the central projection to W with center of projection (,,, ) In the lower part of Z, the retraction r is the projection to W parallel to the diagonal of Z ({(z, z,, z) z [0, ]}) Note that, (i) r({} [0, ) [0, ] [0, ]) p j= W j r([0, ] {} [0, ) [0, ]) p j= W j 2 r([0, ) [0, ] [0, ] {}) p j= W p j (ii) Let z Z such that z has < n p coordinates equal to If z W then r(z) = z, ie, r(z) has all the coordinates equal to coordinates of z If z Z W then z belongs to the top of Z and we can assume, without loss of generality, that z = (,,, x n+,, x p ) Thus, r(z) = (,,, x n+,, x p ) + λ z(,, ) = (,,,, x n+ + λ (x n+ ),, x p + λ (x p )) W, Therefore, the coordinates in z that are equal to remain equal to in r(z) W Using the retraction r, we define a retraction ρ : Y π (W ) by ρ([a, u ],, [a p, u p ]) = ([a, u ],, [a p, u p]), ([a, u ],, [a p, u p ]) Y, where (u,, u p) = r(u,, u p ) We have that ρ is continuous and from (i) and (ii), it follows that ρ is well-defined Now, we shall show that s r = r s First, we observe that s(w ) W Let P Z such that P belongs to the bottom of Z We take the vector v = (,,, ) Then, (s r)(p ) = s(r(p )) = s(p + λ v), for some λ R such that P + λ v W Thus, s(w ) W (s r)(p ) = s(p + λ v) = s(p ) + λ v = r(s(p )) = (r s)(p ) Now, let P Z such that P belongs to the top of Z Let u = P (,,, ) and u = s(p )(,,, ) Then, (s r)(p ) = s(r(p )) = s(p + λ u), for some λ R and such that P + λ u intersects W Then, (s r)(p ) = s(p + λ u) = s(p ) + λ u s(w ) W = r(s(p )) = (r s)(p )

8 8 EDIVALDO L DOS SANTOS AND FRANCIELLE R DE C COELHO Therefore, s r = r s Thus, s p r = r s p Let us consider σ = σ π (W ) We have that σ (π (W )) π (W ) and σ generates a free Z p -action on π (W ) Claim: ρ : (Y, σ) (π (W ), σ ) is a Z p -equivariant map Indeed, (σ ρ)([a, u ],, [a p, u p ]) = σ ([a, u ],, [a p, u p]) where (u,, u p) = r(u,, u p ) = ([a p, u p], [a, u ],, [a p, u p ]), (ρ σ)([a, u ],, [a p, u p ]) = ρ([a p, u p ], [a, u ],, [a p, u p ]) with (ũ p, ũ,, ũ p ) = r(u p, u,, u p ) Now, we have that On the other hand = ([a p, ũ p ], [a, ũ ],, [a p, ũ p ]), s p (ũ p, ũ,, ũ p ) = s p 2 (ũ p, ũ p, ũ,, ũ p 2 ) = s(ũ 2, ũ 3,, ũ p, ũ ) = (ũ,, ũ p ) s p (ũ p, ũ,, ũ p ) = s p (r(u p, u,, u p )) s p r=r s p = r(s p (u p, u,, u p )) = r(u,, u p ) Then, (ũ,, ũ p ) = r(u,, u p ) and thus, (ũ,, ũ p ) = (u,, u p) Therefore, ρ σ = σ ρ and then, ρ is a Z p -equivariant map From this, we conclude that gen (Y, Z p ) gen (π (W ), Z p ) Note that W = ( p i= W i) ( p j= W j ) ( p j= W p j ) is written as an union of p 2 closed subsets of Z By simplicity, we rewrite W = p2 i= W i Let W = { 2} [ 2, ] [ 2, ] and we compute π (W ) : π (W ) = {([a, u ],, [a p, u p ]) Y π([a, u ],, [a p, u p ]) W } Then, = {([a, u ],, [a p, u p ]) Y (u,, u p ) W } ( A { 2 3 = } ) A [0, ] A [0, ] Y

9 COINCIDENCE THEOREMS FOR MAPS OF FREE Z p-spaces 9 dim π (W ) (k ) + (p )k = p k For each W i, i = 2, 3,, p 2, p 2, in the analogous way, we obtain that dim π (W i ) p k, i = 2, 3,, p 2, p 2 p 2 p 2 Since π (W ) = π = π (W i ) is an union of closed subsets i= W i i= with dim π (W i ) p k, i =, 2,, p 2, by [4, The Sum Theorem], it follows that Then, by Theorem 27, Therefore, dim π (W ) p k gen (π (W ), Z p ) dim π (W ) + (p k ) + = p k gen (Y, Z p ) gen (π (W ), Z p ) p k, which completes the proof of lemma Now, suppose that f : X Y has no Z p -coincidence points As in the proof of Theorem (a), there is a Z p -equivariant map φ : X Y Then, it follows from Lemma 3, that gen(x, Z p ) gen(y, Z p ) p k, which contradicts gen(x, Z p ) n + = pk + This completes the proof of Theorem (b) (Case (c) n < pk and gen(x, Z p ) = n + ) Proof In this case, we have that gen(x, Z p ) = n + pk and, it follows from Theorem 26 that there is a Z p -equivariant map F : X S pk Z p, where = Z p Z p Z p is the pk-fold join (Definition 25) On the other hand, it follows from [2, Corollary 6] that there are a Z p -space X, a cone CW-complex Y of dimension k and a map ϕ : X Y without Z p - S pk Z p coincidence points Further, there is a Z p -equivariant map E : S pk Z p X and, consequently, the map f = ϕ E F : X Y has no Z p -coincidence points We observe that this construction shows that the hypothesis n p k in Theorem (a) and (b) is the best condition to guarantee the existence of Z p -coincidence points, when we consider any Hausdorff paracompact Z p - space X of gen(x, Z p ) = n + This completes the proof of Theorem

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