All Questions
8
questions
7
votes
1
answer
482
views
Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...
- 12.8k
8
votes
0
answers
802
views
Second homotopy group of a topological group
It is well-known that any Lie group $G$ has $\pi_2(G)=0$: see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of ...
- 3,126
9
votes
1
answer
646
views
Must an inverse limit of simply connected groups be simply connected?
While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
- 6,823
7
votes
4
answers
2k
views
Topological structure of SO(n) as a product
I’m interested in the question for which $n$ the special orthogonal group is homeomorphic to the product
$$ \mathrm{SO}(n) \approx S^{n-1} \times \mathrm{SO}(n-1). $$
Allen Hatcher [1, p. 293 f.] ...
3
votes
1
answer
142
views
What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,030
3
votes
1
answer
379
views
A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$
Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of ...
- 452
6
votes
3
answers
1k
views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
- 17.3k
8
votes
1
answer
723
views
Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions
The Hilbert-Smith conjecture states that
If $G$ is a locally compact group which acts effectively on a connected manifold as a
topological transformation group then is $G$ a Lie group.
It was ...
- 1,414