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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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:
$$BH \to BG \to BK.$$
...
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2answers
50 views
Good book on topological group theory?
I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
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39 views
Which definition is correct?
I have encountered several different definitions of left Haar measure that don't all seem to agree.
The setting I care about is Locally Compact Groups.
The first seems to completely disagree with ...
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1answer
36 views
Haar measure $\tau$-additive?
I'm reading some results from Measure Theory Volume 4 by D.H. Fremlin, and I'm stuck on something.
This is pulled out of one of his lemmas (stated more generally for topological groups):
A Haar ...
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1answer
25 views
Why is the free pro-c-group on an infinite set not the pro-c-completion of the free group?
The set-up is the following: $\mathfrak c$ is a collection of finite groups closed under subgroups, homomorphic images, and extensions. For any group $G$, the pro-$\mathfrak c$-completion $G(\mathfrak ...
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1answer
20 views
Continuous action of topological group and embedding
Let $G$ be a topological group act continuously on a topological space $X$.
Why the continuity of the action of $G$ on $X$ implies that $G$ embedded as
topological group in $S_{X}$. Here $S_{X}$ is ...
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1answer
144 views
subgroup of connected locally compact group
I need a reference or a short proof for the following property:
A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
4
votes
1answer
164 views
Quotient of a locally compact Hausdorff space by a proper action is Hausdorff
I am trying to prove the following:
Let $G$ be a topological group acting properly on a Hausdorff locally
compact space $X$, i.e. preimages of compacts sets by the map
$$G\times X\to X\times ...
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0answers
18 views
a free topological product of topological semigroups?
Much work has be done on describing the topology of free products of topological groups (Graev, Morris, Katz, etc). Could anybody hint me any results on free topological products of topological ...
5
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1answer
192 views
$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.
Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions
$x \mapsto ax$
and
$x\mapsto xa$ are continuous on $G$.
How to prove elementarily ...
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votes
1answer
40 views
Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)
We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer).
For those wondering, we say that a ...
2
votes
1answer
96 views
$G$ acts transitively on connected space, then so does identity component
Suppose $G$ is a topological group that acts on a connected topological space $X$. Show that if this action is transitive (and continuous), then so is the action of the identity component of the ...
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0answers
28 views
The simply-connectedness of quotient space
If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
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0answers
34 views
Example of a finite, non-abelian group in which left invariant metric is also right invariant [duplicate]
I need an example of a finite, non-abelian group $(G, \cdot)$ which satisfies the following condition:
If $d$ is a metric on $G$ such that $d(ax, ay)=d(x,y), \ \ \ \ \forall a,x,y \in G$,
then ...
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0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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1answer
22 views
Factor group of profinite group
Wikipedia (http://en.wikipedia.org/wiki/Profinite_group, Properties and Facts) says that the factor group of a profinite group $G$ by a closed normal subgroup $N$ is another profinite group. No proof ...
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1answer
75 views
On an existence of a quasi-finite left- invariant Borel measure in a non-locally compact Polish group
Let $(G,B(G))$ be a Polish group. A Borel set $A \subset G$ is called Haar null if there is a Borel probability measure $\mu$ in $G$ such that $\mu(g(A))=0$ for each $g \in G$.
A Borel measure ...
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votes
1answer
74 views
A question about quotient under group action
Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ?
If so, can you please ...
3
votes
1answer
100 views
Generating sets for topological groups
Let G be a compact topological group.
Suppose G has a subset X and a normal subgroup N such that the subgroup generated by X is dense in N.
Moreover, suppose G has a subset Y such that the subgroup ...
3
votes
1answer
103 views
Metric on a group
Is there a non-abelian finite group $G$ with the property: If metric $d$ on $G$ is left invariant then is also right invariant?
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1answer
43 views
On compact topological group
Must a compact topological group be metrizable? If not, is it separable?
Thanks for any help.
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1answer
54 views
Cardinality of quotient
Given $X$ a topological space, we consider $\mathcal{F}$ the class of all continuous maps $f:X\to H$ where $H$ is a topological group... (edited) and $|H|\le |X|$
If $f,g\in\mathcal{F}$, say $f:X\to ...
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votes
2answers
29 views
How to show that the circle group T contains a copy of unit interval [0,1]?
Here, $T$ is the set of all complex numbers of absolute value 1. I want to show that there is a (natural) copy of the interval $[0,1]$. Any hint?
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1answer
47 views
An example for a non-precompact minimal topological group.
Do you have an example of a non-precompact minimal topological group?
A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
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1answer
212 views
Every Tychonoff space is an image of a moscow space under a continuous open mapping.
Every Tychonoff space is an image of a moscow space under a continuous open mapping.
A space $X$ is called Moscow if the closure of every open set $U\subset X$ is the union of a family of ...
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0answers
50 views
From positive definite function to Følner sequence -— a question on amenability and nuclearity
We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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votes
3answers
44 views
group of homeomorphisms subgroup
(a) Let X be a topological space. Prove that the set $Homeo(X)$ of homeomorphisms $f:X \to X$ becomes a group when endowed with the binary operation $f \circ g$ .
(b) Let $G$ be a subgroup of ...
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votes
2answers
54 views
Maximal compact subgroups of $GL_n(\mathbb{R})$.
The subgroup $O_n=\{M\in GL_n(\mathbb{R}) | ^tM M = I_n\}$ is closed in $GL_n(\mathbb{R})$ because it's the inverse image of the closed set $\{I_n\}$ by the continuous map $X\mapsto ^tX X$. $O_n$ is ...
5
votes
2answers
119 views
Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?
I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields.
Does anyone know a proof for this result?
3
votes
1answer
143 views
Metrizable group
Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete.
Here is how I am proceeding:
It can be ...
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votes
1answer
137 views
Question about pointwise canonically weakly pseudocompact space.
A point $x$ of a space $X$ is said to be a point of canonical weak pseudocompactness if the following condition is satisfied:
For every canonical open subset $U$ of $X$ such that ...
4
votes
1answer
95 views
Finding an open set for a topological group
Let $G$ be a locally compact topological group, $K$ a compact subgroup and $\Gamma$ a discrete subgroup. I try to find a neighbourhood $U$ of the identity such that $\Gamma \cap UK = \Gamma \cap K$. ...
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1answer
85 views
Is there an example of a non-orientable group manifold?
Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
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votes
1answer
86 views
Noncommutative dual group
If $G$ is a locally compact group, we can define its dual group $\hat G$. That is set of continuous homomorphism from $G$ to circle group $\mathbb T$. My question is how to define dual group $\hat G$ ...
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votes
1answer
63 views
The intersection of open normal subgroups in a compact, totally disconnected topological group is trivial.
I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma.
If a topological group $G$ is compact and totally disconnected, then the open normal ...
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votes
1answer
36 views
Cauchy product on topological rings
Let $R$ be any commutative Hausdorff topological ring. I am looking for a preferably general condition on sequences $(x_n)_{n \in \mathbb{N}}$, $(y_n)_{n \in \mathbb{N}}$ such that the equation $$ ...
3
votes
0answers
102 views
Moscow space-Examples
A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G_{δ}$ -subsets of $X$ .
For example, Every first countable $T_1$ ...
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votes
1answer
40 views
orthogonal group of a quadratic vector space
I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
2
votes
1answer
67 views
fundamental group of a graph
let $G$ be a connected graph and $\Omega$ its universal covering. Let $\gamma_1,\dots,\gamma_r$ be free generators of $\Gamma:=\pi_1(G)$, $v\in\Omega$ be a vertex and $s_i$ a path from $v$ to ...
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votes
0answers
41 views
Is the Hilbert-Smith conjecture still unsolved?
Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then
$G$ is a Lie group.
Is this conjecture still unsolved? Is ...
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1answer
21 views
inverse on topological semigroup
Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$.
Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
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votes
3answers
129 views
what are all the open subgroups of $(\mathbb{R},+)$
I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
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votes
1answer
225 views
Is $\operatorname{Homeo}([0,1])$ Weil-Complete?
After learning about uniformities on topological groups, we were given several sources to read. I came across the term "Weil-complete." A topological group is Weil-complete if it is complete with ...
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votes
1answer
98 views
Why are locally compact groups Weil complete?
Why are locally compact groups Weil complete?
Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent.
Thank you, and sorry if I have bad writing.
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1answer
152 views
Topology induced by the completion of a topological group
Let $G$ be an abelian topological group and let $\hat{G}$ be its completion, i.e. the group containing the equivalence classes of all Cauchy sequences of $G$. What exactly is the topology of ...
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0answers
25 views
Finiteness of fixed points of a Lie group action
Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$.
Under which assumptions there will be a ...
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0answers
50 views
Isometry groups are topological groups (resp. lie groups). Is every topological (resp. Lie-) group an isometry group?
The isometry group of a metric space is a topological group (with the compact open topology).
The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...
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1answer
83 views
Gillman-Jerison Theorem
How can i prove it?
[Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G_{\delta}-dense $ in $X$.
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votes
1answer
61 views
If $H$ and $G/H$ are compact, then $G$ is compact.
Suppose that $G$ is a topological group and that $H$ is a subgroup of $G$ so that $H$ and $G/H$ are compact. I am trying to show that $G$ must be compact.
The first idea is to use the natural map ...
3
votes
1answer
67 views
Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.
$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider
$$
\Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
