Proof of Brouwer's Fixed Point Theorem.

Question

What is the simplest way to prove Brouwer's Fixed Point Theorem in three dimensions?

Answer 1

With homology perhaps: Assume the $3$-disc admits a continous map without fixed points. Tracing the line from $x$ through $f(x)\neq x$ up to the boundary, we get a retraction $r$ from $D^3$ onto the $2$-sphere: $ri=id$. Using the functoriality of homology gives $id_*=r_*i_*$. But this isn't possible: $H_2(S^2)=Z$ while $H_2(D^3)=0$ and identity map can't factor through a trivial space.

Answer 2

The simplest proof (not only for $n=3$, but for every $n$) is the Analysis proof of John Milnor (i.e., no use of Algebraic Topology), and in particular, its simplified version by C.A. Rogers:

1. Milnor, John, Analytic proofs of the "hairy ball theorem'' and the Brouwer fixed-point theorem, Amer. Math. Monthly 85 (1978), no. 7, 521–524.

2. Rogers, C. A., A less strange version of Milnor's proof of Brouwer's fixed-point theorem, Amer. Math. Monthly 87 (1980), no. 7, 525–527.

A simplified version of Rogers' proof can be found here.