Using Brouwer's fixed point theorem

TL;DR

This paper explores Brouwer's fixed point theorem and its implications in topology, combinatorics, and geometry, including applications to the HEX game, interval piercing, and the evasiveness problem, highlighting both classical and advanced results.

Contribution

It presents a comprehensive overview of Brouwer's fixed point theorem's consequences, including new insights into its applications and stronger related theorems.

Findings

Connections between Brouwer's theorem and the HEX game

Results on piercing multiple intervals

Discussion of stronger theorems and their applications

Abstract

Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem.