# 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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103 views

### Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable.
A family $\mathcal F$ of subsets of a topological space ...

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89 views

### Conjugacy in metaplectic groups

Let $F$ be a non-Archimedean local field (characteristic 0) and $G=GL(2,F)$. Let $\tilde{G}$ be "the" metaplectic double cover of $G$ (defined using an explicit cocycle as in Gelbart's book (Weil's ...

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195 views

### Actions of locally compact groups on the hyperfinite $II_1$ factor

Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.
(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
(2) If so, how does one ...

**3**

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**1**answer

87 views

### A converse of Cartan's automatic continuity theorem

Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...

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108 views

### Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...

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69 views

### Extrinsic applications of Haar measure

I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such ...

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202 views

### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

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81 views

### Topological groups with homeomorphic underlying spaces, isomorphic abstract groups and homotopy equivalent classifying spaces

Define the classifying space $BG$ of a well-pointed topological group $G$ as the fat realization of the nerve of $G$.
Let $G$, $H$ be well-pointed topological groups. Assume that there is a ...

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61 views

### Invariant measures on locally compact homogeneous spaces

Edit: answered in the first comment. This turned out to be really easy, as the orbits of an open $\sigma$-compact subgroup yield a partition of $X$ into open $\sigma$-compact subsets.
Let $G$ be a ...

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**1**answer

392 views

### Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...

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130 views

### Failure of Schur's lemma for topological group representations

Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a ...

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40 views

### Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...

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**2**answers

213 views

### Set of topologies on a group making it a compact Hausdorff topological group

Maybe stupid, but from the following well known facts about compact Hausdorff (CH) spaces:
CH topologies on a given set are pairwise incomparible (one is not finer or coarser than the other).
There ...

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**1**answer

272 views

### Model structure on the category of topological groups

Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...

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113 views

### Bi-invariant metrics on compact Polish group

Let $(X,\tau,\circ)$ be a compact Polish group. Is there necessarily a metric $d$ on $X$ inducing $\tau$ such that $d(x \circ a,x \circ b) = d(a \circ x, b \circ x) = d(a,b)$ for all $x,a,b \in X$?
...