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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 $\mathbb{R}^4$s still in any way canonically Lie groups?

   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 $\mathbb{R}^4$s still in any way canonically Lie groups?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 4th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexOf course: as bare groups, $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$ are isomorphic. As for exotic $\mathbb{R}^4$: no, they are not.

   Of course: as bare groups, $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$ are isomorphic.

   As for exotic $\mathbb{R}^4$: no, they are not.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 4th 2010
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   Author: Urs
   Format: MarkdownItex> Of course: as bare groups, $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$ are isomorphic. Ah, of course. Thanks. We identify the underlying sets using the Peano curve and then of course the group structure of $\mathbb{R}$ induces a wild (smoothly speaking) group structure on $\mathbb{R} \times \mathbb{R}$.

   Of course: as bare groups, $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$ are isomorphic.

   Ah, of course. Thanks. We identify the underlying sets using the Peano curve and then of course the group structure of $\mathbb{R}$ induces a wild (smoothly speaking) group structure on $\mathbb{R} \times \mathbb{R}$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexWait, let me see. I am looking for an example (or not) of two different Lie group structures whose underlying bare group structure coincides.

   Wait, let me see. I am looking for an example (or not) of two different Lie group structures whose underlying bare group structure coincides.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeOct 4th 2010
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   Author: Mike Shulman
   Format: MarkdownItexI think Todd meant to consider both $\mathbb{R}$ and $\mathbb{R}\times\mathbb{R}$ with their usual group structures. They are isomorphic because both are vector spaces of dimension $2^{\aleph_0}$ over $\mathbb{Q}$.

   I think Todd meant to consider both $\mathbb{R}$ and $\mathbb{R}\times\mathbb{R}$ with their usual group structures. They are isomorphic because both are vector spaces of dimension $2^{\aleph_0}$ over $\mathbb{Q}$.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, right. Thanks.

   Ah, right. Thanks.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 5th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexaded a remark along these lines to [[Lie group]]

   aded a remark along these lines to Lie group