I was wondering about the following:
Is it possible for a compact, metric, topological group to be locally path-connected but not locally contractible?
Here "locally contractible" means that every point has a local basis of contractible neighborhoods. This is sometimes called "strongly locally contractible."
Originally I also asked specifically about the finite-dimensional case, but in the comments below it's pointed out that it's been solved: Such groups are Lie groups and thus locally contractible in the strong sense. So, an example for groups where this fails would have to be infinite-dimensional. Especially, the Menger Cube is an example of a compact, metric, homogeneous space that's locally path-connected but not locally contractible; thus the full strength of "group" could well be necessary.
I'm asking in relation to this question, to give a bit of motivation. This group isn't even locally compact, and it's known that there are manifolds whose homeomorphism groups aren't locally contractible even in a much weaker sense:
Local Contractibility of Homeomorphism Group of $\mathbb{R}^n$