# Questions tagged [topological-groups]

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### Is there an explicit construction of the Bohr Compactification of the Integers?

Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including ...

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### 6-functor formalism for topological stacks

I am trying to understand the 6-functor formalism of sheaves on topological stacks. As explained in this answer, there is a 6-functor formalism of sheaves for locally compact Hausdorff spaces, which ...

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### Does any locally compact topological group which is not Hausdorff have a Haar measure?

In 'Linear Analysis and Representation Theory' by Steven Gaal at the end of Chapter IV, page 227, the author claims that any locally compact topological group $G$ which is not Hausdorff has a Haar ...

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### Reference for homotopy and homology theory of topological groups

I am looking for references which deal with the homotopy theory and homology theory of general topological groups, not necessarily compact, or anything. I am eyeing towards certain infinite-...

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### Mattuck's Theorem for abelian varieties for a non-locally compact field

Let $A$ be an abelian variety of dimension $d$ defined over a complete ultrametric field $K$ of dimension $0$. Let us put on $A(K)$ the topology induced by the one of $K$ (for example, following ...

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### Subset of an open retract is a retract [migrated]

I have a doubt about how to start thinking the following problem:
** Problem **
Let $X$ be a topological space and $A\subset X$ an open set. If $A$ is a retract of $X$ and $B\subset A$, is $B$ a ...

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### How to construct morphisms of $C^*$ - algebras between $\mathbb{C}$ and $ C_0 (X) \rtimes_r G $?

Let $G$ be a locally compact, Hausdorff, and second countable topological group.
Let $X$ be a proper, second countable $G$ - space with compact quotient $G \backslash X $.
Let $C_0 (X)$ be the $C^*$ - ...

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### Product map on topological group measurable?

Let $G$ be a topological group and $\mathcal{B}$ its Baire $\sigma$-algebra (i.e. the smallest $\sigma$-algebra for which all continuous functions $G\rightarrow\mathbb{R}$ are measurable). Consider ...

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### Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$

Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...

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### What is the universal/fine uniformity on a topological group?

Cross posted from https://math.stackexchange.com/questions/4889335
I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\...

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### Embeds in a topological W-group, or a W-space that embeds in a topological group?

In Theorem 3.11 of Tkachuk - A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace it's shown that if a compact Hausdorff space embeds in a topological W-...

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### $\mathbb{Z}$-homomorphism and $\mathbb{Z}_p$-homomorphism

$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions.
Is $\mathrm{Hom}_{\mathbb{Z}...

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### In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...

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### $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...

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### When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...

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### Image of a complete topological group under open and surjective map is complete?

A uniform space $X$ is complete if every Cauchy filter in $X$ is convergent. Here we do not require $X$ to be Hausdorff.
Question.
Let $G$ be a complete topological group and let $H$ be a topological ...

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### Compact subgroups of a linear group over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...

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### Does every (Abelian) Polish group have a nontrivial locally compact subgroup?

The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...

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### Classifying space of centralizer

$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let
$$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$
be the homotopy ...

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### Are closures of products of unimodular subgroups unimodular?

Let $G$ be a locally compact group, $N \subset G$ a unimodular normal subgroup, and $H \subset G$ a discrete (hence unimodular) subgroup. Does it follow that the closure $\overline{NH} \subset G$ is ...

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### Maximal protori in compact topological groups

I read the following proposition in the article in the link. Since $G$ itself is finite-dimensional, isn't the maximal protorus $T$ also finite-dimensional? In this case, $G$ becomes a Lie group. Isn'...

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### Is the union of 1-dimensional pro-tori in a finite dimensional pro-torus dense?

Is the union of 1-dimensional compact connected abelian subgroups in a finite dimensional compact connected abelian group dense?

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### Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...

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### Using the von Neumann crossed product to introduce a measure on the orbit space?

Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question: is there a natural way of using the ...

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### Is there a purely topological definition of $\text{Spin}(p,q)$?

I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty).
A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...

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### Topologies on diffeomorphisms groups

Suppose that $M$ is a finite-dimensional $C^{\infty}$-manifold, and let $\mathrm{Diff}\left(M\right)$ be the group of $C^{\infty}$-diffeomorphisms from $M$ to itself. When $M$ is compact, the usual ...

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### Can Rep(G) tell us whether G is discrete?

Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...

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### An abelian category with a full embedding from topological abelian groups

I know this is a very vague question, but I can't think of a better question to ask.
Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \...

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### Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...

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### Uniqueness of left-invariant Borel probability measure on compact groups

On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide?
It is classical that the Haar ...

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### Examples of non-discrete, cocompact subgroups

I am looking for non-trivial examples of the following:
$G$ is a locally compact group
$H\subset G$ a closed subgroup
Both are unimodular and non-discrete
The quotient space $G/H$ is compact, but $G$ ...

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### Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]

Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$
i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$
Now let $X$ be the restriction of its ...

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### Generalized conjugacy classes in (topological) groups

Let $G$ be a topological group. We define an equivalence relation on $G$ as follows:
For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate:
$$x\mapsto ax,\qquad x\...

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### Minimal F-semi-norms

There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...

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### (Non)complete abelian groups in the “transfinite p-adic topology”

For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$
$$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...

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### Does $\Bbb Z[X]$ determine $X$?

For a Hausdorff space $X$, consider the free abelian group $\mathbb{Z}[X]$ generated by $X$. Equip it with the finest topology which makes the map $X\to\mathbb{Z}[X]$, $x\mapsto [x]$ continuous and ...

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### Are U(H) and PU(H) locally uniform topological groups with the norm topology? Towards an instance of infinite-dimensional Hilbert's Fifth Problem

In looking at the work of Enflo generalising Hilbert's Fifth Problem from the Euclidean to the Banach case, there are the following conditions:
the multiplication in the topological group is locally ...

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### The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...

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### Idempotent conjecture and non-abelian solenoid

Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...

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### Idempotent conjecture and (weak) connectivity of (a reasonable) dual group

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...

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### Parametrization of topological algebraic objects

There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...

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### What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...

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### $T_1$ paratopological group having a dense commutative subgroup is commutative

I'm learning about topological groups from Arhangelskii and Tkachenko "Topological groups and related structures" and this is one of the exercises there.
A paratopological group is a group ...

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### Non-continuous group homomorphism from p-adic field to C*

Let $F$ be a p-adic field. A character on $F^\times$ is defined as a continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group ...

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### Topologies that turn the real numbers into a compact Hausdorff topological group

If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or ...

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### Continuity of central character [closed]

Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group ...

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### Is norm-continuous representation factored through a Lie quotient group?

I asked this 11 days ago at MSE, but there was no answer, I hope people here could help.
Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...

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### Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...

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### Discreteness of $D^{-1}D$ given that $D$ is uniformly discrete

Let $G$ be a topological group with unit element $e$.
We say that $D\subseteq G$ is discrete if for all $x\in D$ there is a unit-neighborhood $U\subseteq G$ such that $x^{-1}D\cap U=\{e\}$. We say ...

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### Homotopic but not equivariantly homotopic maps

Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to ...