All Questions
Tagged with at.algebraic-topology lie-groups
241
questions
3
votes
0
answers
162
views
Equivalent condition for compact group to act transitively
Let $ M $ be a connected manifold. Let $ \pi_1(M) $ be the fundamental group of $ M $.
Suppose there exists a compact group $ K $ that acts transitively on $ M $. Then $ \pi_1(M) $ must have a finite ...
- 3,991
11
votes
1
answer
441
views
Asking whether there is a compact Lie group containing affine symplectic group
The affine symplectic group is interesting and important in physics. However, the Lie group is noncompact. In order to have some good properties (Basically, we need some good behavior of Haar measure) ...
- 149
3
votes
0
answers
235
views
CW structure on $\mathrm{PU}(3)$/Heisenberg group
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where
$H$ is the Heisenberg group of order 27
$G_{81}$ is the No. 9 group of order 81 (...
- 10.4k
4
votes
0
answers
311
views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.
Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,401
1
vote
0
answers
136
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 ...
- 10.4k
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 ...
- 10.4k
2
votes
0
answers
163
views
Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?
Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
- 10.4k
5
votes
0
answers
194
views
Outer and inner automorphism of $\mathrm{Pin}$ groups
$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
- 317
6
votes
0
answers
147
views
Cohomology of $\mathrm{BPSO}(2d)$ with $Z_2$ coefficients
$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.
$\BPSO(2d)$ is the classifying ...
- 487
4
votes
1
answer
292
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
14
votes
1
answer
339
views
On the homological dimension of a Borel construction
Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$...
- 11.9k
4
votes
0
answers
176
views
Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.4k
5
votes
0
answers
133
views
Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.4k
8
votes
1
answer
250
views
The "canonical fibration" for the Lie group $G_2$
$\DeclareMathOperator\SO{SO}$At the very begining of Akbulut and Kalafat - Algebraic topology of $G_2$ manifolds, the authors stated that there is a "canonical fibration" for $G_2$ of the ...
- 935
1
vote
1
answer
155
views
Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]
Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
- 317