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Questions tagged [topological-groups]

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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Is a continuous map between two topological groups homotopic to a homomorphism between them?

Let $G$ and $H$ be two topological groups and $f:G\to H$ be a continuous map. Is there a continuous homomorphism $g:G\to H$ homotopic to $f$?
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A more Intuitive proof of regularity of topological group [closed]

The proof given in Munkres's Topology is quite unnatural at least to me. He uses the notion of symmetric neighboorhood and I'm pretty sure it's quite ad hoc. After all, is it really useful? Can ...
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1answer
62 views

Yamabe's theorem proof

*I'm trying to make the proof of Yamabe's Theorem that says that an arcwise connected subgroup of a Lie group G is a Lie subgroup of G. I found the proof in Goto's article (https://www.ams.org/...
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33 views

Homomorphism from $p$-adic to $l$-adic groups

I have seen and heard the statement that the $p$-adic and $l$-adic topologies are incompatible. I would appreciate a proof or references supporting this statement. More precisely, I am interested in ...
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1answer
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If $G$ is profinite, and $A$ is discrete, $f: G \to A$ is continuous $\implies$ $f$ factors through a normal open subgroup

Let $G$ be a profinite group; that is compact, and totally disconnected. Take $A$ a discrete space, and a continuous map $f: G \to A$. $\exists N$ open and normal in $G$ and a continuous map $g: G/...
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Simple groups and fundamental groups

What does it mean, topologically, to have a simple fundamental group? For instance, the torus $S^1 \times S^1$ has $\mathbb Z \times \mathbb Z$not simple. The case of $S^1$ is $\mathbb Z$, not simple ...
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Upper-triangular subgroup is not unimodular

Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$. My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not ...
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Is a finitely generated metrizable group discrete?

The question is in the title. A countable locally compact Hausdorff group is discrete, so saying that a finitely generated metrizable group is locally compact would be enough. What if the group is a ...
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1answer
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Is $\Bbb Q / \Bbb Z$ discrete?

I would like to say that $\Bbb Q / \Bbb Z$ is not discrete (when $\Bbb Q$ has euclidean topology), since $\Bbb Z \subset \Bbb Q$ is not open. But OTOH we have $$\Bbb Q / \Bbb Z \cong \bigoplus_p \Bbb ...
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A word of length 4 in a local group, with two different values

I've been told that in a local group, one can find a word of length of $n=4$, i.e $w=g_1 g_2 g_3 g_4$, with two different meanings depending on how we put the parentheses. However I've also read that ...
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Every element $g$ of $G$ has a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$

Let $G$ be a topological group and $g\in G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$. If $g=e$, l have proved it. ...
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No semisimple Lie group acting on Klein bottle

How to show that there is no semisimple Lie group can act transitively on the Klein bottle? So in general, if $X$ is a homogeneous space of a solvable Lie group. Is it true that $X$ can't be written ...
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1answer
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Is $\hat{G}$ is complete with respect to the induced topology of $G$?

For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $\hat{G}$. The induced fundamental system of neighbourhoods of $\...
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Properties of open subsets of the real numbers

Let $G$ be an open subset of $\mathbb{R}$ . Then: a) Is the set $H=\{xy|x,y\in G\ \text{and}\ 0\notin G\}$ open in $\mathbb{R}$? b) Is the set $G=\mathbb{R}$ if $0\in G$ and $\forall x,y\in G, x+y\...
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Proving that Sp(2N,R) is not locally compact

I'm working though Hall's Lie groups, Lie algebras, and representations and I want to show that the matrix Lie group $Sp(2N,\mathbb{R})$ is not locally compact. I've already shown that it fails to be ...
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1answer
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a question about the definition of the Chabauty topology for discrete groups

Let $G$ be a discrete group and let $S(G)$ the set of subgroups equipped with the following topology: A net $(H_i)_i\subset G$ of subgroups converges in the Chabauty topology to a subgroup $H$ of $G$ ...
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24 views

Is this topological transformation group locally path connected?

A surface is an oriented connected sum of $g\geq 0$ tori, with $b \geq 0$ open disks removed, and $n \geq 0$ punctures in its interior. Let Aut$^+(S,\partial S)$ denote the group (under composition) ...
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1answer
27 views

Word norm in compact subsets of finitely generated groups

Let $G$ be a finitely generated topological group, not necessarily discrete. Fix a finite generating set $S$ and denote by $|x|$ the word norm of $x \in G$ with respect to this generating set, i.e., ...
2
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1answer
48 views

Continuity(?) of induced group representation in the isometries of $L^p$

Let $G$ be a topological group which acts on a measure space $(X,\mu)$ by measure preserving transformations. It's well known that this induces a representation $\pi$ of $G$ in $O(L^p (X,\mu))$, the ...
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1answer
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Proof of the existence of liftings in Introduction to Algebraic Topology by Rotman

The following proof is given in Introduction to Algebraic Topology by Rotman Now I understand the proof except for the following line Since $S^1$ is a multiplicative group there is a telescoping ...
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Topologies on isometry groups

Let $X$ be a (complete) Hilbert space, not necessarily separable. What are the known interesting topologies on the group $Isom(X)$ of its isometries? Are there interesting metrics on this group?
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Locally compact topological group is paracompact

Let $G$ be a locally compact, connected topological group.Show that $G$ is paracompact.
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CW approximation of a topological group

Suppose we have a topologiacal group $G$. How do we construct a CW-complex $\bar{G}$ which is also a topological group, such that $\bar{G}\to G$ is a weak equivalence?
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Group Actions on $\Bbb R^n$

We are learning about Quotient spaces, and group actions. For any $n\in\mathbb{N}$. We know that the function $G(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=1\}$ is a group action on $\mathbb{R}^n$. I am ...
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Quotient map from topological group onto left cosets is open

I've been trying to prove the following exercise from "James Munkers'-a first course in topology": Let $G$ be a topological group, and let $H\leq G$. Induce the left cosets, $G/H$, with the quotient ...
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1answer
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Every group of totally disconnected type is locally profinite?

Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type. On the other hand, we ...
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1answer
38 views

Is every totally disconnected topological group locally profinite?

Let $G$ be a topological group which is totally disconnected. Then one point sets in $G$ are closed, and hence $G$ is Hausdorff. On the other hand, we have a notion of a locally profinite group, a ...
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1answer
47 views

Weak containment of trivial representation

Let $\sigma$ be a continuous unitary representation of the topological group $G$ on a Hilbert space $V$. Suppose $\sigma$ weakly contains the trivial representation, that is: for any compact subset $K$...
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Extension of a finite group by a connected group necessarily splits?

Suppose that $G$ is a compact abelian group. I denote by $G_0$ the connected component of the identity in $G$. If $G_0$ is open in $G$ (equivalently $G/G_0$ is finite) is it true that $G\cong G_0\...
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1answer
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Transformation group of a topologic space problem

I'm reading Steenrod The topology of fiber bundle and at the page 7 is this statment that I don't understand. Namely the last proposition. Take $\phi: G\to\text{Aut}(Y)$ given by $\phi(g)(y)=gy.$ If ...
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1answer
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Give examples of two $C_2$ actions on $S^n$ such that the orbit spaces are not homotopy equivalent.

Give examples of two $C_2$ actions on $S^n$ such that the orbit spaces are not homotopy equivalent. My attempt: For $n \ge 2$. I considered the following actions of $C_2$ on $S^n$ : (i) mapping to ...
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0answers
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Reference request for locally compact groups proof

Where can I find a proof of the following: "Every locally compact group is a directed union of $\sigma$-compact open subgroups" This is claimed in Greenleaf's book but I haven't been able to find a ...
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1answer
63 views

If a covering space of a topological space X has a topological group structure, when we transfer this structure on X?

Let (G,e) be a topological group and p : G → X be a covering map. When p can transfer the group structure to make X a topological group? It is clear that if p is a group homomorphism it is done. But ...
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group homomorphism from a profinite group continuous iff kernel open

I have a question regarding a (probably simple) fact. However I am lacking some basic topological knowledge. Let $G$ be a locally pro finite group, i.e. ever open neighborhood of $1_G$ contains a ...
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Locally compact nilpotent group has an open subgroup isomorphic to $\mathbb{R}^n\times K$

My question is about a possible generalization of the following structure theorem of locally compact abelian groups. Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact ...
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1answer
23 views

Continuity of absolute value in topological ordered abelian groups

Let $(G,+,0)$ be an abelian topological ordered group, that is, $G$ is endowed with a total order $\leq$ such that, for any $a,b,c\in G$, we have that $a\leq b$ implies $a+c\leq b+c$. Moreover, $G$ is ...
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1answer
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$(X/H)/(G/H)=X/G$? Here $G$ is a topological group acting continuously on $X$, $H$ is a closed normal subgroup of $G$.

Let $G$ be a topological group acting continuously on a topological space $X$ (on the left, denoted by $g\cdot x$, $g\in G, x\in X$), $H$ be a closed normal subgroup of $G$. Then The quotient ...
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Definition of the Fell topology: Completion with respect to a seminorm

I'm reading about the Fell topology and have a question on some preliminary material. My reference is these notes on automorphic representations. Let $G$ be a locally compact Hausdorff, second ...
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Covering of a topological group

Let $X$ be a topological group with a locally path-connected, path-connected covering $(\tilde{X},p)$. If we fix an $u\in p^{-1}(e)$, we should be able to deduce a unique group structure for $H=\tilde{...
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Is every topological groupoid equivalent to a disjoint union of topological groups?

It's a fact that any groupoid is equivalent to a disjoint union of (deloopings of) groups. See, e.g. Proposition 4.3 of https://ncatlab.org/nlab/show/groupoid#PropertiesEquivalencesOfGroupoids. Does ...
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1answer
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Every point in completion of a topological group in closure of a countable subset

Is there a name or perhaps some interesting equivalent condition for the following condition on an abelian topological group $G$ with uniformity generated by the neighbourhoods of 0? Every point ...
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1answer
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What is $SO(n+1)/O(n)$ as a topological space?

Consider $SO(n),O(n)$ as topological groups. Find out $SO(n+1)/O(n)$ as a topological space. My attempt: Observed the inclusion : $O(n) \hookrightarrow{} SO(n+1)$ by, $$A \mapsto\begin{bmatrix} det(...
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1answer
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Prove that $[0,1]^n / S_n \cong \Delta^n$

Prove that $[0,1]^n / S_n \cong \Delta^n$, ( $S_n$ denotes the permutation group on n symbols, $\Delta^n:=\{(x_0,\dots,x_n)\in\Bbb R^{n+1} : x_0,\dots,x_n \ge 0 , x_0+\dots+x_n=1\}$, and the action is ...
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1answer
29 views

“Halving” neighborhoods in topological groups

Let $S=B\left(0,\frac{\epsilon}{2}\right)$ be a ball in $\mathbb{R}^n$.  Given a translate $S+t$, there is the nice property that, for all $x,y\in S+t$, we have $x-y\in S-S\subseteq B(0,\epsilon)$...
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1answer
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Do dense topological subgroups always have the same finite quotients as their underlying bigger topological subgroup?

Definition: Let $G$ be a topological group. We call $G$ a finite quotient of $G$ if there exists a normal subgroup $N$ of $G$ such that $H = G/N$ and $H$ is finite. Let $G'$ be a dense subgroup of $G$...
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Hilbert's 5th problem and short exact sequences of Lie groups.

Let $G$ be a locally compact topological group. Suppose that there exists a normal closed subgroup $H$ which is also a Lie group such that $G/H$ with the quotient topology is also a Lie group. Prove ...
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1answer
55 views

Compactness of $PGL(n, \mathbb{R})$ and $PGL(n, \mathbb{C})$

I know that the $PGL(n, \mathbb{R})$ or $PGL(n, \mathbb{C})$ is Lie group, because $PGL(n, F) = GL(n, F) / Z(n, F)$, where $Z(n, F)$ - scalar transformation and $F$ is $\mathbb{C}$ or $\mathbb{R}$. ...
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1answer
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Connected components of $PGL(n, \mathbb{R})$ ond $PGL(n, \mathbb{C})$

I know that the $PGL(n, \mathbb{R})$ or $PGL(n, \mathbb{C})$ is Lie group, because $PGL(n, F) = GL(n, F) / Z(n, F)$, where $Z(n, F)$ - scalar transformation and $F$ is $\mathbb{C}$ or $\mathbb{R}$. ...
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0answers
21 views

Unique characterizing property of identity component

Is there any characterizing property of path component of identity in a topological group? In other words, how will I show that some path connected subgroup is the identity path component of ...
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1answer
39 views

All connected closed subgroups in $\rm{SO}(3)$

I want to find all connected closed subgroups in $\rm{SO}(3)$. Is there a direct classification without attracting Lie algebras? Thank you very much!