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

Abstract

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 $f:X\to M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^{\ast }(V_k)=0$, for $k\ge 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that $\mathrm{ind}X\ge n>(|G|-r)m$, 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