# Tagged Questions

53 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 ...
58 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 ...
65 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 ...
706 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 ...
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 ...
113 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 ...
105 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 ...
127 views

### Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
46 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 ...
51 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 ...
102 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 ...
112 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 ...
163 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.
175 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: $$BH \to BG \to BK.$$ ...
90 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 ...
186 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 ...
219 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 ...
136 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 ...
74 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 ...
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 ...
131 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 $properly\;discontinuous$ ...
### 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 ...