Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The idea is like this: consider $T = \{ (x,y,z) \mid x + y + z = 1, x, y, z \geq 0 \}$. This is a triangle in $\mathbb R^3$. Take a point $\overline x \in T$, and consider $\lambda_x M \overline x \in T$, for some $\lambda_x \in \mathbb R$. This is a vector which is equal to some $y \in T$. In particular, $\lambda_x M$ is a homeomorphism $T \to T$. Hence it has a fixed point $\overline x$. So $\lambda_x M \overline x =\overline x \implies M \overline x = \frac{1}{\lambda_x} \overline x$. So $\overline x$ is an eigenvector with eigenvalue $\frac{1}{\lambda _x}$, which is certainly positive.
I was extremely surprised when this question came up, as we are studying fundamental groups at the moment and that doesn't seem the least bit related to eigenvalues at first. My question is: what are some other example of surprising applications of topology?
Best Answer
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.