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.
 A: Brouwer's fixed point theorem can be illustrated using the game Hex.  In fact, the determinacy of Hex is equivalent to the 2-dimensional version.
A: Beginning students might appreciate applications outside mathematics. Here are a couple of current (undergraduate) projects: The topology of data and 3D image analysis. The tool used is called "persistent homology".
A: The Jordan-Brouwer separation theorem is a nice application of singular homology.
A: You can prove the Grushko decomposition theorem, as well as the Kurosh subgroup theorem.  See for example the last two sections of Massey's "Algebraic Topology: an Introduction".
Steve
A: So I just stumbled across a very cool, easy to read paper by Eckmann called "Social Choice and Topology A Case of Pure and Applied Mathematics." It's a survey of a result that Eckmann proved a long time ago about spaces that have "n-means," and the application of this result to social theory. Basically, if you have some function that assigns a single preference when given the preferences of a population, and this function is "continuous" and fair (you can switch the order of the population and it won't change the output, and if everyone has the same preference then that's what you get out), then this says a LOT about the space of preferences. Given a few assumptions about the space of preferences (I dunno how reasonable they are or realistic, since I have no idea what it would mean for a space of preferences to be a cell complex...), one can show right off the bat that the fundamental group is abelian (we have an H-space structure), and later on he basically classifies all possible spaces that admit social choice functions. Anyway, I thought this was pretty cool as an example of beautiful mathematics and a quirky "application."
A: Here is one interesting application of Algebraic Topology in Molecular Biology:
Connolly, M.L. An application of algebraic topology to solid modeling in molecular biology. The Visual Computer 3, 72–81 (1987). https://doi.org/10.1007/BF02153663
For a computer related application please see this link:   http://www.lix.polytechnique.fr/~sanjeevi/atmcs/
A: Kakutani's fixed point theorem allowed John Nash to prove his crucial result that started the whole field of game theory in like 9 pages or something -- a very sexy maths thesis.
Brouwer's theorem appears in economics as well in "satisficing" solutions (behavior based on choosing "better" options rather than "the best"), for example:
Becker, Robert A.; Chakrabarti, Subir K., Satisficing behavior, Brouwer’s fixed point theorem and Nash equilibrium, Econ. Theory 26, No. 1, 63–83 (2005). Zbl 1097.91005.
And, Brouwer's fixed-point theorem was used by Arrow & Debreu to establish general equilibrium theory. That theory is widely despised but mathematically beautiful, however nowadays it's considered more believable if your paper says "I rely on no fixed-point theorems".
A: 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. 
A: 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. 
A: The only finite-dimensional division algebras over $\mathbb{R}$ have dimensions 1, 2, 4, and 8.  Note that we do not require that there be an identity element. For weaker versions of this result, see pages 173 and 222 in Hatcher's book Algebraic Topology.
Edit: An earlier version of this answer stated, mistakenly, that the only finite-dimensional division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$.  For this to be true, one needs the additional hypothesis that there is an identity element.
A: This isn't a specific application but an area of application.  Robert Ghrist and others use algebraic topology as a way of integrating local data about sensor networks into global information.  For example, you may want to determine whether there are any holes in your sensor coverage.  Here's an interview with Ghrist where he elaborates on this idea.
A: How about the ham sandwich theorem? Or is that Ham and Cheese? 
There is a plane which cuts all 3 regions into equal parts.
