my favorite algebraic topology fact:
a topological group is a topological space with a group structure, ie an associative binary operation, an identity element, and inverses.

if X is a topological group with identity i, then the fundamental group pi_1(X,i) is always abelian

if there is interest i’ll do a thread on why this is the case. this is one of those things that i found genuinely surprising the first time i read it and still surprising when i went through the proof.

ok let's go!

the first thing that you might be thinking is that it's not terribly surprising that abelian groups show up somewhere with fundamental groups. after all, all the higher homotopy groups are abelian!

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so when i first learned it, the proof was given as an exercise, broken down into four parts:
for a topological group X with identity i, let \\Omega(X,i) be the set of loops in X based at i. You can define a binary operation on the loop space by multiplying "pointwise".

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