All Questions
Tagged with at.algebraic-topology lie-groups
42
questions
148
votes
7
answers
22k
views
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
- 3,994
18
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 $...
- 773
13
votes
4
answers
4k
views
Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
- 14.6k
23
votes
6
answers
5k
views
cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.3k
20
votes
2
answers
1k
views
The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
- 19.1k
13
votes
3
answers
2k
views
Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
- 2,694
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
2
votes
0
answers
425
views
A Generalized De Rham cohomology
Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by $L_{\mathbb{C}}^{...
- 452
51
votes
2
answers
2k
views
$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group
Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...
- 42.7k
41
votes
3
answers
3k
views
What is the classifying space of "G-bundles with connections"
Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
- 53.5k
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 62.3k
31
votes
2
answers
1k
views
Is Lie group cohomology determined by restriction to finite subgroups?
Consider the restriction of the group cohomology $H^*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of ...
- 413
27
votes
5
answers
3k
views
Is there a Morse theory proof of the Bruhat decomposition?
Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
- 8,098
26
votes
2
answers
5k
views
Cohomology of Lie groups and Lie algebras
The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
- 23.4k