Examples where geometry/topology helps algebra I think that the most beautiful parts of mathematics are the ones where algebra and geometry/topology interact with each other. For example, in algebraic topology we often transform geometric problems in algebraic ones, so we could say that in this case "algebra helps geometry". Are there examples the other way around?
I know, for example, that given a commutative ring $R$, we can functorially construct the topological space $\text{Spec}(R)$. But does studying the spectrum through topological methods actually offer algebraic results on $R$ that were harder to find out in a purely algebraic manner? Is it useful in this sense?
More in general, I would like to find out examples where geometry/topology helps algebra.
 A: Another less elementary example is the  $(1,2,4,8)$-theorem on division algebras. Here topological methods are very helpful for the proof, e.g., using Bott periodicity.
A: For proving $a^2+b^2=c^2$ for right triangles, "geometry" can be very helpful:

A: Here's a favorite example. It's quite tedious to prove using algebraic means the useful fact that

Proposition. Any subgroup of a free group on finitely many letters is free.

On the other hand, one can prove it using topology as follows. Let $\Gamma$ be a subgroup of the free group on $n$ letters, $F_n$. A rose with $n$ petals has fundamental group $F_n$. By the covering space correspondence, there is a cover $X$ of the rose with $n$ petals with $\pi_1(X) \cong \Gamma$. As $X$ covers a graph $X$ is itself a graph, and so must have free fundamental group.
A: Gromov proved his theorem on groups of polynomial growth very geometrically.
The theorem states that a group $G$ has a nilpotent subgroup of finite index (a very algebraic property) if and only if $G$ has polynomial growth (a fundamentally geometric property.)
It arguably kicked off the entire area of geometric group theory.
