Applications of algebraic topology What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the better. All the above can be proved with just the fundamental group. More involved applications would be nice.

How about the ham sandwich theorem? Or is that Ham and Cheese? 
There is a plane which cuts all 3 regions into equal parts.
