All Questions
Tagged with topological-groups at.algebraic-topology
17 questions with no upvoted or accepted answers
10
votes
0
answers
199
views
"Homotopy homomorphisms" of homeomorphisms of Euclidean space
For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
- 8,167
10
votes
0
answers
435
views
The group analogue of the James construction
If $(X,e)$ is a based topological space, then the James construction on $(X,e)$ is the free topological monoid with unit $e$: $J(X)=\coprod_{n\geq 1}X^n /\sim $ where $(x_1,...x_{j-1},e,x_j,...,x_n)\...
- 7,193
8
votes
0
answers
819
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 ...
- 3,146
7
votes
0
answers
149
views
Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
- 10.5k
7
votes
0
answers
315
views
Two different Thom diagonals in recent literature?
Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...
- 10.4k
5
votes
0
answers
192
views
When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?
The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...
- 1,174
4
votes
0
answers
147
views
Does the self-homeomorphism group of a finite CW complex have CW homotopy type?
Let $X$ be a finite CW complex and form the group $\mathcal{H}(X)$ of self-homeomorphisms $X\xrightarrow{\cong}X$, furnishing it with the compact-open topology. Under the assumptions on our space $\...
- 5,596
4
votes
0
answers
205
views
Why is any $G$-resolution a principal $G$-bundle?
In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a ...
- 79
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
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 61
3
votes
0
answers
151
views
Reference for homotopy and homology theory of topological groups
I am looking for references which deal with the homotopy theory and homology theory of general topological groups, not necessarily compact, or anything. I am eyeing towards certain infinite-...
- 219
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
2
votes
0
answers
193
views
A $\mathbb{Z}_2$-equivariant map from $n$-torus to $2$-sphere that is null-homotopic is $\mathbb{Z}_2$-homotopic to a non-surjective map?
I have been thinking on the problem below for a while and I am not sure if it is correct or not. I am trying to see if there exists a counter-example for the problem below.
Problem: Let $f: (S^1)^n \...
- 21
2
votes
0
answers
191
views
Factoring a topological universal cover
Let $X$ be a compact, connected, locally path-connected, and semilocally simply connected topological group with $\pi_1(X) \cong \mathbb{Z}$.
Let $u\colon \widetilde{X}\to X$ be its topological ...
- 998
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
1
vote
0
answers
132
views
Nilpotency of topological groups
A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...
- 901
1
vote
0
answers
49
views
Weighted cancellation norm of a word computation
A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$.
We say that two disjoint pairs $\{...
- 111