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I guess I should know this, but when asked today I couldn’t give a good answer:
What can we say about putting two inequivalent structures of a Lie group on a given bare group?
A maybe related basic question: are the exotic ℝ4s still in any way canonically Lie groups?
Of course: as bare groups, ℝ and ℝ×ℝ are isomorphic.
As for exotic ℝ4: no, they are not.
Of course: as bare groups, ℝ and ℝ×ℝ are isomorphic.
Ah, of course. Thanks. We identify the underlying sets using the Peano curve and then of course the group structure of ℝ induces a wild (smoothly speaking) group structure on ℝ×ℝ.
Wait, let me see. I am looking for an example (or not) of two different Lie group structures whose underlying bare group structure coincides.
I think Todd meant to consider both ℝ and ℝ×ℝ with their usual group structures. They are isomorphic because both are vector spaces of dimension 2ℵ0 over ℚ.
Ah, right. Thanks.
aded a remark along these lines to Lie group
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