All Questions
Tagged with at.algebraic-topology lie-groups
241
questions
4
votes
2
answers
383
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 389
4
votes
0
answers
159
views
Is there any results about the stable (or unstable) cohomology operations on cohomology of Lie groups?
$\DeclareMathOperator\SU{SU}$For the $\mod p$ singular cohomology of classical Lie groups, such as $H^*(\SU(n); \mathbb{Z}/p\mathbb{Z})$, there are well known results about the actions of the stable ...
- 71
4
votes
1
answer
230
views
Homotopy groups of quotient of SU(n)
Let $X$ be the quotient topological space obtained by identifying the matrices $A$ and $\overline{A}$ in the topological group $\mathrm{SU}(n)$ (here $\overline{A}$ denotes entry-wise complex ...
- 271
3
votes
0
answers
139
views
Equivariant classifying space and manifold models
The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
- 793
4
votes
1
answer
405
views
Faithful locally free circle actions on a torus must be free?
Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...
- 375
7
votes
2
answers
479
views
Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
- 125
6
votes
1
answer
550
views
Cobordism cohomology of Lie groups
Are there any results about cobordism cohomology of Lie groups？For example, $\mathrm{MU}^*(\mathrm{SU}(n))$.
- 61
4
votes
0
answers
123
views
Real Representation ring of $U(n)$ and the adjoint representation
I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
user340793
10
votes
0
answers
291
views
Compact Lie groups are rational homotopy equivalent to a product of spheres
According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
- 30.8k
10
votes
2
answers
614
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 ...
- 6,684
6
votes
1
answer
579
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 3,991
5
votes
0
answers
129
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 3,991
9
votes
1
answer
407
views
Compact flat orientable 3 manifolds and mapping tori
There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are ...
- 3,991
21
votes
1
answer
820
views
What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?
Consider the following partial order. The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m =...
- 1,025
2
votes
1
answer
428
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 3,991