Topological Tverberg Theorem: the proofs and the counterexamples

TL;DR

This paper reviews the history, key constructions, and properties related to the Topological Tverberg Theorem, including detailed analysis of the classifying space for permutation groups and clarifies bibliographical issues.

Contribution

It provides a comprehensive overview of the theorem's proofs, counterexamples, and the cell structure of associated classifying spaces, clarifying complex topological concepts.

Findings

Detailed description of the classifying space $K(S_r,1)$

Analysis of important constructions and their properties

Clarification of bibliographical issues in the field

Abstract

I describe the history of Topological Tverberg Theorem. I present some important constructions and discuss their properties. In particular, I describe in details the cell structure of the classifying space $K\left( S_{r},1\right),$ where $S_{r}$ is the permutation group. I also clarify some bibliographical issues.