Configuration-like spaces and coincidences of maps on orbits

TL;DR

This paper investigates configuration-like spaces of points in Euclidean space avoiding k-wise coincidences, providing new upper bounds on their genus and deriving theorems related to coincidence points of continuous maps.

Contribution

It introduces new upper bounds on the genus of configuration-like spaces under group actions and extends Cohen--Lusk type theorems for coincidence points.

Findings

New upper bounds on the genus of configuration-like spaces

Generalizations of Cohen--Lusk type theorems for Euclidean maps

Insights into the structure of spaces with group symmetries

Abstract

In this paper we study the spaces of $q$-tuples of points in a Euclidean space, without $k$-wise coincidences (configuration-like spaces). A transitive group action by permuting these points is considered, and some new upper bounds on the genus (in the sense of Krasnosel'skii--Schwarz and Clapp--Puppe) for this action are given. Some theorems of Cohen--Lusk type for coincidence points of continuous maps to Euclidean spaces are deduced.