All Questions
Tagged with topological-groups at.algebraic-topology
7 questions
23
votes
5
answers
4k
views
Fundamental groups of topological groups.
Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a non-...
- 273
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
16
votes
2
answers
1k
views
rationalization of classifying spaces
This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an $H$-...
- 7,637
5
votes
1
answer
443
views
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$
I am interested in knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(3)$ or $BO(3)$.
Here we can take either:
$B^...
- 2,147
4
votes
0
answers
133
views
Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 404
3
votes
0
answers
490
views
Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$
I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
- 21.5k