$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$
| Autor: | Taciana O. Souza, Denise de Mattos, Edivaldo L. dos Santos |
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| Rok vydání: | 2017 |
| Předmět: |
General Mathematics
Natural number Type (model theory) Cohomological dimension Topology 52A35 01 natural sciences Prime (order theory) Integer 55M35 FOS: Mathematics Algebraic Topology (math.AT) Mathematics - Algebraic Topology 0101 mathematics $(H G)$-coincidence 55S35 Mathematics Finite group Conjecture Primary 55M20 52A35 Secondary 55M35 55S35 Group (mathematics) topological Tverberg theorem 010102 general mathematics 55M20 010101 applied mathematics $G$-action |
| Zdroj: | Bull. Belg. Math. Soc. Simon Stevin 24, no. 4 (2017), 567-579 |
| ISSN: | 1370-1444 |
| Popis: | 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 $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that ${\rm{ind}}\, X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. 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 $r$. 13 pages |
| Databáze: | OpenAIRE |
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