Measuring the size of the coincidence set

Gonçalves, Daciberg Lima; Jaworowski, Jan; Pergher, Pedro L. Q.

doi:10.1016/s0166-8641(01)00292-9

Cited by 11 publications

(15 citation statements)

References 5 publications

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“…The following theorem is a version for manifolds of the main result in [8] Proof. Since n > (|G| − r)m ≥ m, f * (V k ) = 0, for all k ≥ 1.…”

Section: Let Us Consider the Inclusionmentioning

confidence: 95%

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Mattos¹,

Santos²,

Souza³

2017

Bull. Belg. Math. Soc. Simon Stevin

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Let X be a paracompact space, let G be a finite group acting freely on X and let H a cyclic subgroup of G of prime order p. Let f :In this work, we estimate the cohomological dimension of the set A(f, H, G) of (H, G)-coincidence points of f . Also, we estimate the index of a (H, G)coincidence set in the case that H is a p-torus subgroup of a particular group G and as application we prove a topological Tverberg type theorem for any natural number r. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently, fail for all r that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any natural number r. main directions considered of this problem are either when the target space Y is a manifold or Y is a CW complex. In the first direction are the papers of Borsuk [4] ( the classical theorem of Borsuk-Ulam, for H = G = Z 2 , X = S n and Y = R n ), Conner and Floyd [5] (for H = G = Z 2 and Y a n-manifold), Munkholm [13] (for H = G = Z p , X = S n and Y = R m ), Nakaoka [14] (for H = G = Z p , X under certain (co)homological conditions and Y a m-manifold) and the following more general version proved by Volovikov [17] using the index of a free Z p -space X (ind X, see Definition 2.2 ):Theorem A.[17, Theorem 1.2] Let X be a paracompact free Z p -space of ind X ≥ n, and f : X → M a continuous mapping of X into an m-dimensional connected manifold M (orientable if p > 2). Assume that:

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“…The following theorem is a version for manifolds of the main result in [8] Proof. Since n > (|G| − r)m ≥ m, f * (V k ) = 0, for all k ≥ 1.…”

Section: Let Us Consider the Inclusionmentioning

confidence: 95%

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Mattos¹,

Santos²,

Souza³

2017

Bull. Belg. Math. Soc. Simon Stevin

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“…We note that similar methods can be used to estimate sizes of "(H, G)-coincidence sets" (see [11] for a definition), as exhibited in [16].…”

Section: Proof (1) It Is Clear That Propertiesmentioning

confidence: 99%

General Bourgin–Yang theorems

Błaszczyk¹,

Marzantowicz²,

Singh³

2018

Topology and its Applications

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“…Gonçalves, Jaworowski and Pergher [3] have defined (H, G)-concidence for a continuous map f from a n-sphere S n into a k-dimensional CW-complex Y , where G is a finite group which acts freely on S n and have proved that if H is a nontrivial normal cyclic subgroup of a prime order, then…”

Section: Introductionmentioning

confidence: 99%

On nonsymmetric theorems for $(H,G)$-coincidences

Mattos¹,

Santos²

2009

TMNA

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Let X be a compact Hausdorff space, ϕ: X → S n a continuous map into the n-sphere S n that induces a nonzero homomorphism ϕ * : H n (S n ; Zp) → H n (X; Zp), Y a k-dimensional CW-complex and f : X → Y a continuous map. Let G a finite group which acts freely on S n. Suppose that H ⊂ G is a normal cyclic subgroup of a prime order. In this paper, we define and we estimate the cohomological dimension of the set Aϕ(f, H, G) of (H, G)-coincidence points of f relative to ϕ.

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Measuring the size of the coincidence set

Cited by 11 publications

References 5 publications

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

General Bourgin–Yang theorems

On nonsymmetric theorems for $(H,G)$-coincidences

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