All Questions
16 questions
19
votes
2
answers
1k
views
Exotic smooth structures on Lie groups?
If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $...
- 783
10
votes
2
answers
697
views
Homotopy properties of Lie groups
Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example
$G$ is homotopy equivalent to a smooth compact ...
- 6,962
10
votes
1
answer
732
views
What's the sufficient or necessary conditions for a manifold to have Lie group structure?
For example, given a Lie group, its fundamental group must be Abelian. So $\Sigma_g$ ($g>1$) can't have Lie group structure. We also know for $S^n$ only $n=0,1,3$ can have Lie group structures.
...
- 249
8
votes
1
answer
228
views
Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at ...
- 10.5k
7
votes
1
answer
490
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{\...
- 13.6k
7
votes
2
answers
728
views
Euler class of S^1-orbibundle
Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^...
- 73
5
votes
0
answers
133
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 311
4
votes
1
answer
304
views
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$
Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts.
1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
- 317
3
votes
1
answer
347
views
Homotopy type of $SO(4)/SO(2)$
A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What ...
- 325
3
votes
0
answers
134
views
When do quotients of $G$-vector bundles exist?
Let's work in the category of smooth (paracompact, Hausdorff) manifolds. Let $M$ be a manifold and $G$ a Lie group acting on $M$. Suppose $E$ is a $G$-vector bundle on $M$ (that is, $G$ acts on $E$ by ...
- 51
2
votes
1
answer
257
views
How to extend an equivariant map from a compact Lie group
Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$...
- 1,040
2
votes
0
answers
98
views
Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support
Let $(Z,A)$ a compact ENR pair, then
$$\chi(Z)=\chi_c(Z-A)+\chi(A)$$
where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
- 795
2
votes
0
answers
71
views
Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147
1
vote
0
answers
138
views
Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
- 10.5k
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
0
votes
0
answers
88
views
Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]
Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
- 10.5k