Surprising applications of topology 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?

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