Are all elements of an abelian fundamental group path homotopic?

Question

Consider the fundamental group of some topological space $X$. Is it true that if this group is abelian, then all elements are necessarily path homotopic?

Answer 1

I have a slightly silly answer:

Remember the fundamental group relies on a choice of basepoint. So if you take any disconnected space (in particular not path connected) "the" fundamental group will only refer to one of the components. And thus can be abelian when the space is disconnected.

This is one place where the fundamental groupoid begins to look more appealing, as the fundamental groupoid of a disjoint union of spaces is the disjoint union of the groupoids.

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For a less silly example, you can run the same argument as outlined in the answer to this question. But take $W$ to be your favorite path connected space with a trivial fundamental group.

Again, this relies on the fact that you need a choice of basepoint. But this isn't really cheating, because the fundamental group can't detect things in other path components (since it's constructed with paths). So any counterexample to claims involving path-disconnected spaces will need to be at least a little bit silly in this way.

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I hope this helps ^_^