1
vote
1answer
34 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
1
vote
1answer
80 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
3
votes
1answer
61 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
2
votes
0answers
78 views

How often is a torus in a compact Lie group nullhomologous?

Minor nomenclature question: What is the standard name for the ring structure induced on the homology of an H-space by its multiplication? I've seen "homology ring" and "Pontrjagin ring." Hopefully ...
4
votes
2answers
60 views

$\pi_n(SU(2)/Z_N)\simeq?$, $\pi_n(SO(3)/Z_N)\simeq?$, $\pi_n(U(1)/Z_N)\simeq?$

So based on the tool, I have attempted to compute the following threes homotopy groups. $\pi_n(SU(2)/Z_N)\simeq?$ $\pi_n(SO(3)/Z_N)\simeq?$ $\pi_n(U(1)/Z_N)\simeq?$ With a fixed positive integer ...
5
votes
1answer
142 views

Computing characteristic numbers of homogeneous spaces

I apologize in advance if this is a bad question. I would like to prove or refute a conjecture about the vanishing of characteristic numbers of homogeneous spaces, and to this end am looking (and ...
3
votes
1answer
95 views

What is the induced map on fundamental group of the inclusion of unitary group in the orthogonal group?

What is the induced map on fundamental group of the inclusion of unitary group $U(n)$ in the orthogonal group $SO(2n)$?(Note that the unitary group $U(n)$ can only embedded in the group $SO(2n)$, not ...
10
votes
1answer
111 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
1
vote
0answers
36 views

Group of continuous automorphisms of cylindrical plane is a lie group

How do I prove this theorem? Theorem: The group of continuous automorphisms of a cylindrical plane is a lie group. In this context, cylindrical means the Laguerre plane. I found a paper it ...
5
votes
0answers
85 views

$\pi_2(G)$ for $G$ a Lie group. [duplicate]

It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory? Also, who gave the first proof of ...
1
vote
0answers
102 views

Cohomology ring of $U(n)$

As you know $$H^\ast (U(n);{\bf Z})=\bigwedge_{\bf Z}[x_1,x_3,...,x_{2n-1}]$$ where $|x_i|=i$ To prove this we use Leray-Hirsch Theorem for $$\tag{*}\ U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$$ ...
0
votes
1answer
55 views

A question about double cover of Lie group

If the fundamental group of a symplrctic Lie group be infinite cyclic, why it should has a unique connected double cover?
1
vote
1answer
69 views

Contractibility vs. G-contractibility

Let $X$ be a space equipped with an action of a compact Lie group $G$. Recall that such a space is said to be $G$-contractible if the identity map of $X$ is $G$-homotopic (i.e., homotopic through ...
2
votes
0answers
62 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
73 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
191 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 ...
6
votes
1answer
349 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
36 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
135 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
278 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 ...
3
votes
0answers
85 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 ...
2
votes
1answer
330 views

fundamental group of $U(n)$

Is my logic is correct? $f:U(n)\rightarrow U(1)$ defined by $f(A)=detA$ is a group homomorphism so that induces $f^{*}: \pi_1(U(n))\rightarrow \pi_1(U(1))$ will be an Isomorphism right(I am not ...
5
votes
2answers
331 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 ...
4
votes
1answer
324 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
347 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
375 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 ...
21
votes
2answers
302 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
138 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 ...
10
votes
5answers
328 views

$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$

I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$. At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
3
votes
0answers
56 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 ...
7
votes
1answer
543 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 ...
21
votes
1answer
458 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
145 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
81 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
262 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 ...
17
votes
2answers
2k 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
694 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 ...