TITLE: Contractible topological groups
QUESTION [14 upvotes]: Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?

REPLY [17 votes]: If a topological group is contractible, then it is locally contractible (using the group operation produce a contraction which does not move the unit of the group). By a classical result of [A. Gleason, R. Palais, On a class of transformation groups, Amer. J. Math.
79 (1957), 631–648], a locally path-connected finite-dimensional topological group is a Lie group and being contractible, is homeomorphic to an Euclidean space.