0
votes
0answers
3 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
1
vote
2answers
55 views

if $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?

Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) If $\pi_1(G)$ is ...
1
vote
0answers
17 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
4
votes
1answer
54 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
3
votes
2answers
73 views

Showing that $\pi(G/H, 1) = H$ under a condition

Problem: Let $G$ be a simply connected (i.e., $\pi(G)=1$) topological group, and let $H$ be a discrete normal subgroup. Prove that $\pi(G/H,1) = H$. I know that since $H$ is a discrete subgroup ...
2
votes
1answer
75 views

A question on the classifying space $BG$, its universal property (?), and the stack $[\bullet/G]$

I am learning about the classifying space $BG$ of a topological group $G$. I know the definition $$BG=EG/G,$$ where $EG$ is any contractible space on which $G$ acts freely. If I am not mistaken, with ...
6
votes
4answers
746 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
0
votes
1answer
57 views

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
3
votes
2answers
135 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
2
votes
2answers
188 views

The fundamental group of a topological group is abelian

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I ...
7
votes
2answers
135 views

Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
1
vote
0answers
48 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
1
vote
0answers
55 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
3
votes
0answers
107 views

Discrete Closed Subgroup H of a Simply Connected Topological Group G isomorphic to fundamental group of G / H.

A problem in Rotman's Algebraic Topology is as follows: Given a simply connected topological group G with a closed discete normal subgroup H, show that $\pi_1(G / H) \cong H$. I believe I have this ...
10
votes
1answer
115 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 ...
11
votes
1answer
164 views

Is there a nontrivial topological group that's isomorphic to its fundamental group?

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement. Thanks in advance.
4
votes
1answer
203 views

A short exact sequence of groups and their classifying spaces

Suppose that we have a short exact sequence of topological groups: $$1 \to H \to G \to K \to 1.$$ I have found some papers mentioning that the above sequence induces a fibration: $$BH \to BG \to BK.$$ ...
2
votes
1answer
92 views

fundamental group of a graph

let $G$ be a connected graph and $\Omega$ its universal covering. Let $\gamma_1,\dots,\gamma_r$ be free generators of $\Gamma:=\pi_1(G)$, $v\in\Omega$ be a vertex and $s_i$ a path from $v$ to ...
1
vote
2answers
210 views

How is the general linear group a topological group?

How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix ...
4
votes
1answer
233 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 ...
3
votes
2answers
137 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 ...
1
vote
1answer
76 views

A new group operation on the fudamental group

Suppose $G$ is a topological group with operation $\cdot$ and identity element $x_0$. Let $\Omega (G, x_0)$ denote the set of all loops in $G$ based at $x_0$. For $f, g\in\Omega (G, x_0)$ define ...
1
vote
1answer
62 views

$\overline H$ is a normal subgroup of a topological group $G$.

Let $G$ be a topological group. How can we prove that if $H$ is a normal subgroup of $G$, then $\overline H$ is a normal subgroup of $G$ also? First of all, we have to prove that $\overline H$ is a ...
2
votes
1answer
136 views

If the action of a group $G$ on $\mathbb{R}$ is properly discontinuous then G is isomorph to $\mathbb{Z}$?

Let $G$ be a topological group, acts on a topological space $X$, such that the map $f: G \times X \rightarrow X:(g,x)\mapsto g*x$ is continuous. We say that this action is $properly\;discontinuous$ ...
0
votes
0answers
85 views

G is a topological group acts on topological space $X$, is $f_{g}:X\rightarrow X, x\rightarrow g*x$ continuous?

Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$. I am trying to find if $f$ is continuous? my best ...
0
votes
1answer
63 views

$G$ finite group acts freely on top. sp. $X$, can we find for every $x\in X$ an open neighborhood such that:

Let $G$ be finite topological group, and acts freely over the hausdorff topological space $X$, i want to prove that every element $x$ in $X$ has an open neighborhood $U_x$ such that: $g\star ...
6
votes
1answer
98 views

Topological structure of a quotient of ${\rm{SU}}(2)\times{\rm{SU}}(2)$

I'm trying to understand the topology of the product of two three dimensional spheres $\mathbb{S}^3\times \mathbb{S}^3$ quotiented by the action of $\pm 1$ sending a pair of points $(x,y)$ to the ...
4
votes
2answers
120 views

open subsets in topological groups

I'm starting to study topological groups, and I noticed that Every single theorem in topological groups I have to use the following statement: Let $G$ be a topological group and U an open subset of ...
3
votes
0answers
243 views

Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
9
votes
1answer
158 views

Group structure on $\mathbb R P^n$

For which positive integers $n$ can $\mathbb R P^n$ be given the structure of a topological group? I believe that $\mathbb R P^n$ cannot be given a Lie group structure for even $n$, since then it is ...
1
vote
1answer
397 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
3
votes
1answer
85 views

String and BString

In one of the talks of J.P. May he mentioned some examples of structure groups and their classification spaces (he mentioned: O, U, SO, SU, Sp, Spin, String, Top, STop, F and SF). Most of them are ...
8
votes
2answers
227 views

Topological rings which are manifolds

Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume ...
3
votes
1answer
187 views

Does the Sorgenfrey Line have a group operation compatible with its order topology?

The title is the question, but let me explain. Let $\mathbb{L}$ denote the Sorgenfrey line. I and a friend were trying to develop some of the properties of the sorgenfrey line. (if it's metrizable, ...
21
votes
1answer
463 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 $\rho : G\times X\rightarrow X$ ...
7
votes
2answers
762 views

Why is every discrete subgroup of a Hausdorff group closed?

I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together. I have read a useful property of discrete group on the ...
8
votes
2answers
847 views

Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...