# Category Theory: September 5

### This week's topics

We weren't able to get a critical mass for today's class, but here is what I had planned to cover. I'm happy to go over this with anyone later if you are interested.

Poincaré's theorem that the 2-sphere is not homeomorphic to the 2-torus.

This is a classic, standard theorem that should appear in any book that covers homotopy theory. A personal favorite is Munkres, Topology (Second Edition), chapter 9.

Brouwer's Fixed Point Theorem

The proof I was to use is Riehl, Category Theory in Context, Theorem 1.3.3. A similar proof, but using homology (which is easier to generalize to higher dimensions), is in Rotman, An Introduction to Algebraic Topology, chapter 0 (yes, zero). Another proof, using homotopy and vector fields, is in Munkres (Theorem 55.6).