TITLE: Could someone explain me the significance of Brouwer's Fixed Point Theorem?
QUESTION [6 upvotes]: Well, I've been reading the proof of Sperner's Lemma and its use in proving Brouwer's Fixed Point Theorem (including the Coffee stiffing analogy ;-)). But I fail to understand the significance of this theorem. Could someone explain me how it can be applied in other important applications?

REPLY [5 votes]: The number of fixed points of a map is one of the easiest topological invariants to define, yet it holds a surprising amount of topological information. For instance, if $X$ is a manifold and $f$ is a a continuous function of $X$ into itself which is obtained by slightly disturbing the identity map of $X$, then the number of fixed points (counted with appropriate multiplicities) is equal to the Euler characteristic $\chi(X)$. For instance, an infinitesimal rotation of the sphere has two fixed points, so the sphere has Euler characteristic $2$. On the other hand, the same argument shows that the Euler characteristic of the torus is $0$. If $X$ is a Lie group, an infinitesimal translation has no fixed points, so $\chi(X)=0$. Thus, this gives us a very concrete, hands-on way of understanding the Euler characteristic, and we get lots of interesting results seemingly for free. Brouwer's result is the first in a long line of many.