Surprising applications of topology

Solution 1:

Francis Su described in 1999 ("Rental Harmony: Sperner's Lemma in Fair Division", Amer. Math. Monthly, 106, 1999, 930-42) how to apply Sperner's Lemma---which says that every so-called Sperner coloring of a triangulation of an $n$-simplex contains a cell colored with a complete set of colors---to produce a list of variously sized rents for rooms in a shared house that are fair in a certain sense that accounts for all roommates' preferences. See the column "To Divide the Rent, Start With a Triangle", (New York Times 2014 April 28) for an interesting interactive tool that illustrates the algorithm that exploits the lemma.

Solution 2:

How about Furstenberg's proof of the infinitude of prime numbers?

Solution 3:

The proof of the Cayley–Hamilton theorem in the case of different eigenvalues is very easy. The extension to general case in any field is possible using the Zariski topology.

Solution 4:

The Nielsen–Schreier theorem (a subgroup of a free group is itself free) can be proven using methods of elementary algebraic topology.

My personal favourite is the pretty deep result that every finite dimensional divison algebra over $\mathbf{R}$ has dimension $1,2,4$, or $8$. This result seems to be due Kervaire and Milnor; the proof uses methods of advanced algebraic topology. An accessible one-page proof using K- theory was given by Adams and Atiyah, see here (they actually prove a more general result concerning the hopf invariant one problem, of which the statement I mention is a corollary). At present no purely algebraic proof is known.

Solution 5:

The Necklace Splitting problem in combinatorics has been beautifully solved by Alon and West using the Borsuk-Ulam theorem.

http://en.wikipedia.org/wiki/Necklace_splitting_problem