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Feb
25 |
comment |
Is a left topological group which is a manifold a topological group?
That the idea does not work "as is" with $H=K$, where I was looking for a non-continuous $H\rightarrow{}K$ that could then be composed with the canonical $K\rightarrow{}Inn(K)$. But actually I just realized that this can work very easily with $S^1$ and $SO(2)$, considering an isomorphical embedding $S^1\rightarrow{}SO(2)$ and a non-continuous homomorphism $S^1\rightarrow{}S^1$ (which exists). In this way we easily have a non-continuous homomorphism $S^1\rightarrow{}Aut(SO(2))$, and a similar example (that is also compact). |
Feb
24 |
comment |
Is a left topological group which is a manifold a topological group?
It is interesting to point out that the same example cannot work with compact semisimple Lie groups (replacing $xe^{f(y)}$ with the conjugacy $yxy^{-1}$, that sends as well $G\rightarrow{}Inn(G)\subseteq{}Aut(G)$), because in this case any automorphism is automatically continuous, as pointed out in mathoverflow.net/a/40700/3680 |
Feb
24 |
revised |
Is a left topological group which is a manifold a topological group?
added 111 characters in body |
Feb
24 |
comment |
Is a left topological group which is a manifold a topological group?
Nice example, thanks! |
Feb
24 |
accepted | Is a left topological group which is a manifold a topological group? |
Feb
23 |
comment |
Is a left topological group which is a manifold a topological group?
yes, let's also assume $G$ paracompact, with countable atlas as topological manifold. @GeraldEdgar nice theorem, even if I don't see an immediate application to the present case, what is a reference for it? |
Feb
22 |
awarded | Nice Question |
Feb
22 |
revised |
Is a left topological group which is a manifold a topological group?
added 195 characters in body |
Feb
22 |
comment |
Is a left topological group which is a manifold a topological group?
I see... thanks, if you write it as an answer I will accept it. In any case, I was not thinking about this kind of examples, I think I should add that $G$ is supposed to be connected. |
Feb
22 |
comment |
Is a left topological group which is a manifold a topological group?
If you make the coset of a a one-parameter subgroup open (and hence also closed), how can the group still be a topological manifold? |
Feb
21 |
asked | Is a left topological group which is a manifold a topological group? |
Jan
27 |
answered | Ergodicity of composition with a rotation |
Oct
29 |
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Oct
11 |
comment |
A Differential Equation with Nested Functions
can you provide some background? |
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