Tagged Questions

2
votes
0answers
43 views

Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
1
vote
1answer
63 views

Connection between spaces of cosets and homotopy cofibers

Suppose we have an inclusion of a closed sub-Lie group $H\to G$ and take the space of left cosets of $H$, $G/H$. How is this related to the homotopy cofiber of the inclusion of topological spaces ...
6
votes
1answer
150 views

Connectedness of the orthogonal subgroup $O^+_+(k,l)$

Let $O(k,l)$ be the orthogonal group associated to the quadratic form $q$ on $\mathbb{R}^{k+l}$ with signature $(k,l)$. Let $O^+_+(k,l)$ be the connected component of the identity, i.e. the connected ...
2
votes
1answer
135 views

proof by nuke of the fact that fundamental group of topological group is abelian

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
5
votes
1answer
142 views

The Gram-Schmidt process is a deformation retraction

Consider the Gram-Schmidt process $r : GL(n) \rightarrow O(n)$ that sends invertible matrices to orthogonal matrices. I need to show this is a deformation retraction and, by restrictions of $r$, ...
1
vote
1answer
29 views

maximal tori and principal $N(T)$-bundles.

Let $U(n)$ be the unitary group and $T^{n}= S^{1} \times \cdots \times S^{1}$ a maximal torus in $U(n)$. Let $N(T^{n})$ be the normalizer in $U(n)$ of $T^{n}$. How can i prove that $U(n) \rightarrow ...
3
votes
2answers
104 views

why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
15
votes
1answer
189 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
2
votes
0answers
50 views

How to prove that a lie group is simply connected

I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...
3
votes
2answers
202 views

deformation retract of $GL_n^{+}(\mathbb{R})$

Well, I need a deformation retract from $GL_n^{+}(\mathbb{R})$ to $SO(n)$ Here is what I tried, let $A\in GL_n^{+}(\mathbb{R})$ $A=(A_1,\dots,A_n)$ where $A_i$'s are collumn vectors, Recall that the ...
3
votes
1answer
183 views

fundamental group of $GL^{+}_n(\mathbb{R})$

I would like to know whether the $GL^{+}_n(\mathbb{R})$ the set of all invertible matrices with positive determinant is simply connected or not? I guess it is not simply connected but that is just a ...
3
votes
2answers
182 views

Fundamental group of $SO(3)$

How can I show that the universal cover of $SO(n)$, for $n\ge 3$, is a double cover? And how does that reflect the fact that the fundamental group of $SO(n)$ has two elements? What is the relation ...
3
votes
2answers
295 views

$SO(3)$ Lie group

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant ...
16
votes
1answer
221 views

How can I lift a path to $\mathrm{Spin}(n)$?

Suppose I am given an explicit differentiable path $\gamma\colon[a,b]\to SO(n)$, with $\gamma(a)=\gamma(b)=I$. Then $\gamma$ either does or does not lift to a closed loop in $\mathrm{Spin}(n)$. How ...
6
votes
0answers
107 views

Finding the universal cover of a matrix group

Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
3
votes
0answers
55 views

An unexplained iso between $H^{m+1}(O(m+2)/O(m), S^m)$ and $H^{m+1}(S^{m+1})$

I am reading topology of Lie groups by Mimura and Toda and got to the part where they are beginning to compute $H^*(O(n))$, page 120. If we let $r_m :S^m \to O(m+1)$ be the map that sends $v$ to the ...
6
votes
1answer
321 views

De Rham Cohomology of a Lie group

If $G$ is a connected Lie group with Lie algebra $\mathcal{G}$, then de Rham cohomology of left invariant différential forms $H_L^*(G)$ is isomorphic to the Chevalley–Eilenberg cohomology ...
14
votes
0answers
359 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
4
votes
0answers
127 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...
1
vote
0answers
77 views

$X_G$ is a CW complex

I know the following result: If $X$ is a compact smooth manifold and $G$ is a compact Lie group which acts smoothly on $X$, then $X_G = (X\times EG)/G$ is a CW complex. I don't know how to ...
8
votes
1answer
225 views

What is $\pi_i(GL(n))$?

For some reason, I can't find a reference for $\pi_i GL(n,\mathbb C)$ nor can I figure what they are. For most Lie groups, you can get a nice fibration and use the long exact sequence in homotopy to ...
15
votes
2answers
1k views

Visualizing the fundamental group of SO(3)

Recently I became interested in trying to visualize the fact that $\pi_1(\text{SO}(3)) = \mathbb{Z}/2\mathbb{Z}$. For whatever reason, the plate trick doesn't do it for me, so I've been looking for ...
8
votes
2answers
511 views

Homology and Euler characteristics of the classical Lie groups

I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques ...