Is there a nontrivial topological group that's isomorphic to its fundamental group?

algebraic-topology group-theory topological-groups fundamental-groups

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement.

Thanks in advance.


An answer to this can be found on MathOverflow.

The answer on MathOverflow gives an example of a topological group $G$ which is isomorphic to its own fundamental group (as an abstract group). The example is a product of infinitely many $\mathbb{RP}^\infty$'s, or rather a group structure associated to $\mathbb{RP}^\infty$ (or, easier to see, its universal cover $S^\infty$). The underlying group is a vector space over $\mathbb{F}_2$ of dimension $2^{\aleph_0}$.

(Note: This question is lying pretty high up the unanswered group theory question list, so I have added in this community wiki answer to get it off of the list.)

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