TITLE: Surprising applications of topology
QUESTION [33 upvotes]: 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?

REPLY [2 votes]: Here is a nice application in the "real world" of engineering. The use of industrial robots is standard in many manufacturing processes, for example to produce cars. Robot arms resemble a human arm and are able to move to (more or less arbitrary) spatial positions. However, there is the big problem of singular positions. See for example https://blog.robotiq.com/why-singularities-can-ruin-your-day. To overcome this problem, engineers increased the number of joints to gain more flexibility. But to their surprise it was not successful: Singular positions proved to be persistent.
In fact, topology shows that it is impossible to avoid singular positions by increasing the number of joints. This was proved in
Gottlieb, Daniel H. "Topology and the robot arm." Acta Applicandae Mathematica 11.2 (1988): 117-121.