Hilbert’s fifth problem and Gleason metrics

The easy version of this question is: when a topological group is a Lie group?

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Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

Theorem 1 (Hilbert’s fifth problem) Let $latex {G}&fg=000000$ be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then $latex {G}&fg=000000$ is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem) Let $latex {G}&fg=000000$ be a locally compact group, and let $latex {U}&fg=000000$ be an open neighbourhood of the identity in $latex…

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