Could someone explain me the significance of Brouwer's Fixed Point Theorem? 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?
 A: 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.
A: Gérard Debreu and Kenneth Arrow used it to prove some results which led to a Nobel prize in economics. I think that is pretty significant in and of itself.
