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.

This is my favorite. One can show that for any continuous map from $S^{1}$ to $R^{3}$ there is a direction along which the map has at least 4 extrema (in particular, at least 2 global minima and 2 global maxima.) More colloquially, one can show that every potato chip can be placed on a table so its edge touches the table in at least two points and its edge simultaneously has two points of maximum height.

One can prove the fundamental theorem of algebra using the fundamental group of the circle.

Also, as a generalization of the Hairy Ball Theorem, one may compute, for all spheres, the maximum number of linearly independent vector fields that may live on that sphere. (the hairy ball theorem says that this number is 0 for even dimensional spheres, and at least 1 for odd-dimensional spheres). For dimensions less than 15 (I think), one can compute this number using only cohomology operations. The general result was proven by Adams, but I'm not sure how he did it.

As a side note: I don't think you can prove the general versions of Borsuk-Ulam, Brouwer, etc. with just the fundamental group... You need either higher homotopy groups or higher homology groups.