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      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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      2answers
      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 ...
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      votes
      1answer
      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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      votes
      1answer
      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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      1answer
      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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      votes
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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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      votes
      0answers
      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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      2answers
      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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      votes
      1answer
      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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      votes
      3answers
      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 $x,a,b \in X$? ...

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