Browder's Theorem through Brouwer's Fixed Point Theorem

Abstract

One of the conclusions of Browder (1960) is a parametric version of Brouwer's Fixed Point Theorem, stating that for every continuous function $f:([0,1]\times X)\to X$, where $X$ is a simplex in a Euclidean space, the set of fixed points of $f$, namely, the set $\{(t,x)\in [0,1]\times X:f(t,x)=x\}$, has a connected component whose projection on the first coordinate is $[0,1]$. Browder's (1960) proof relies on the theory of the fixed point index. We provide an alternative proof to Browder's result using Brouwer's Fixed Point Theorem.