## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 24, Number 4 (2017), 567-579.

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

Denise de Mattos, Edivaldo L. dos Santos, and Taciana O. Souza

#### 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 $f:X\rightarrow M$ 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 $\ind X\geq n> (|G|-r)m$, 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 integer $r$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 24, Number 4 (2017), 567-579.

**Dates**

First available in Project Euclid: 4 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1515035007

**Mathematical Reviews number (MathSciNet)**

MR3743262

**Zentralblatt MATH identifier**

06848701

**Subjects**

Primary: 55M20: Fixed points and coincidences [See also 54H25] 52A35: Helly-type theorems and geometric transversal theory

Secondary: 55M35: Finite groups of transformations (including Smith theory) [See also 57S17] 55S35: Obstruction theory

**Keywords**

$(H,G)$-coincidence $G$-action topological Tverberg theorem

#### Citation

de Mattos, Denise; dos Santos, Edivaldo L.; Souza, Taciana O. $(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$. Bull. Belg. Math. Soc. Simon Stevin 24 (2017), no. 4, 567--579. https://projecteuclid.org/euclid.bbms/1515035007