Unexpected use of topology in proofs

One day I was reading an article on the infinitude of prime numbers in the Proof Wiki. The article introduced a proof that used only topology to prove the infinitude of primes, and I found it very interesting and satisfying. I'm wondering, if there are similar proofs that use topology where it's not obvious that it can be applied. I'm sure that seeing such proofs could also strengthen my intuition with topology.

So my question is: "Which theorems, not directly linked with topology, have interesting proofs that use topology?".

Thanks in advance!

Solution 1:

One of the richest sources for this is geometric group theory, and especially the parts of it that construct topological spaces corresponding to particular groups (schreier graphs, cayley complexes, BG spaces etc.) For example, the proof that any subgroup of a free group is free (Nielsen Schreier) epitomizes how topological arguments greatly simplify some parts of group theory.

There are proof methods for theorems from discrete geometry that utilize methods from equivariant topology (such as the ham-sandwich theorem via Borsuk ulam.) These use the CS/TM paradigm, which studies maps between the space of "geometric configurations" to the "Space of viable solutions."

See here for a general reference and wikipedia for a shorter exposition.

There are some other miscellaneous examples that I know of:

Bezout's identity via separating curves (assumes the euclidian algorithm in disguise.)
Fundamental Theorem of Algebra (resp. existence of eigenvalues) via lefschetz fixed point theorem
the use of "Denseness arguments" in algebraic geometry (such as cayley-hamilton).

Solution 2:

How about the Nielsen-Schreier theorem? The statement of the theorem is entirely algebraic:

Every subgroup of a free group is free.

To prove it, we use algebraic topology: the free group on $n$ generators is the fundamental group of the bouquet of $n$ circles, and any subgroup of the fundamental group is the fundamental group of some covering space of this bouquet. Then it's a matter of using topological arguments to show that every covering space of the bouquet of $n$ circles is itself homotopy equivalent to some bouquet of circles.