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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.

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    $\begingroup$ The solution of Fermat's last theorem is such an example. $\endgroup$ Jun 27 at 11:48
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    $\begingroup$ Quillen’s revolutionary work used homotopy theory to give the correct definition of the full K-theory of rings. $\endgroup$
    – Randall
    Jun 27 at 11:49
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    $\begingroup$ The topology of $\mathrm{Spec}(R)$, and other spectral spaces, is definitely powerful but mostly from a point-set topology perspective -- not from a truly 'topological perspective' as I am imagining you are after. $\endgroup$ Jun 27 at 12:25
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    $\begingroup$ This is a highly related question which essentially asks the same thing except "group theory" instead of "algebra." $\endgroup$
    – rschwieb
    Jun 27 at 14:11
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    $\begingroup$ Geometric group theory fits the bill. $\endgroup$ Jun 27 at 14:15

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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.

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  • $\begingroup$ Another result related to this is that, you can embed all $F_n$'s inside $F_2$ (by finding an $n$-fold covering of $S^1 \vee S^1$). In fact, you can also embed $F_\infty$ (free group on countably many generators) inside $F_2$. This is quite counter intuitive because the analogous result doesn't hold for abelian group (or more generally abelian categories). You cannot embed a free module of rank $3$ inside a free module of rank $2$. $\endgroup$
    – feynhat
    Jun 30 at 1:25
  • $\begingroup$ I wish I could give you and Dietrich Burde the "best answer", because both your answers were amazing. Unfortunately, I had to choose! $\endgroup$
    – Kandinskij
    Jul 23 at 6:31
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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.

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For proving $a^2+b^2=c^2$ for right triangles, "geometry" can be very helpful:

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  • $\begingroup$ @WarlockofFiretopMountain This is one of the most common proofs. $\endgroup$ Jun 27 at 15:08
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    $\begingroup$ I don’t think that this is an interaction between algebra and geometry. Pythagora’s theorem is a purely geometric result and the drawing doesn’t constitute a formal proof, but a neat geometric visualization. Where is algebra exactly here? $\endgroup$
    – Kandinskij
    Jun 27 at 16:51
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    $\begingroup$ For me the algebra is in the formula $a^2+b^2=c^2$, where I also think of Pythagorean triples and algebraic number theory, and of $a^n+b^n=c^n$. For Pythagorean triples geometry is indeed helpful, to find a parametrisation $a=m^{2}-n^{2},\ \,b=2mn,\ \,c=m^{2}+n^{2}$. Perhaps this would be a better example. $\endgroup$ Jun 27 at 19:42
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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.

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