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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Bigon question related to Dehn twists

Perhaps someone can help me with this: For simple closed curves on an orientable compact surface, if they form a bigon, then is it true that at the intersection points the orientations must be ...
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305 views

Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-1}$ ...
2
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1answer
96 views

Intuition on characters of topological groups

I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't ...
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2answers
68 views

what does linear type mean?

What does it mean when we say a topological group $\Gamma$ has linear type? Is it an algebraic property or a topology property? I wonder if anyone could give some references.
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185 views

Why is that $\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$?

$\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$, that is, every character of $\mathbb{R}/\mathbb{Z}$ is of the form $x\mapsto e(mx)$ for some integer $m$. I was considering the dual of ...
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1answer
93 views

existence of infinite abelian subgroup in infinite locally compact groups

1) Let $G$ be an infinite locally compact group. Does there exist an infinite abelian locally compact subgroup of $G$? Rem: I know that there exists an infinite abelian subgroup in every infinite ...
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1answer
85 views

Proving that an embedding $G \hookrightarrow BC(G)$ is continuous.

I'm going over Professor Tao's presentation of the Birkhoff-Kakutani theorem and I don't see how it follows that $j = (g \mapsto \tau_gf)$ (between "Lemma 2" and "Remark 2") is continuous. I don't ...
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1answer
595 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
3
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1answer
187 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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2answers
387 views

Component of identity

Could you please help me to solve this one: The connected component of the identity of a topological group is a normal subgroup? I also need a hint to show path-connected components are normal ...
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6answers
474 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
6
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1answer
237 views

A compact Lie group has descending chain condition on closed subgroups.

Proposition: Let $G$ be a compact Lie group and let $$G\supset G_1\supset G_2\supset\ldots$$ be a chain of closed subgroups $G_i$ of $G$. Then this chain must eventually stabilize. Question: The hint ...
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1answer
95 views

Topological fields questions

From Wikipedia: "Let $K$ be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications $K$ will be either the field of ...
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2answers
1k views

About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this ...
7
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1answer
379 views

Question regarding the quotient space.

After recently learning about uniformities on topological groups, I've been looking at various problems. I'm having trouble with the following: If $H$ is a closed subgroup of a topological group $G$, ...
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1answer
135 views

Weil's proof of a theorem on finite irreducible representations of products of compact groups

Theorem Let $G$ and $H$ be compact groups. Let $ρ$ be a finite dimensional irreducible continuous representation of $G×H$ over the field of complex numbers. Then $ρ$ is a tensor product of irreducible ...
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1answer
242 views

If both $H$ and $G/H$ are locally compact then $G$ is locally compact (topological Group)

How do I prove this statement? Let $G$ be a Topological group and let $H$ be a subgroup of $G$, if both $H$ and $G/H$ are locally compact then $G$ is locally compact. (we will endow the set $G/H$ ...
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1answer
688 views

The group of invertible linear operators on a Banach space

Let $X$ be a Banach space. Let $G$ be the group of invertible linear operators from $X$ to itself. Now my questions are: If $G$ is equipped with the operator norm topology, how do you show that it ...
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1answer
2k views

The special orthogonal group is a manifold

How can we show that $SO(n)$ is an $n^2$-manifold. It would be tempting to say that $SO(n)$ is an open set of $\mathbb R^{n^2}$ but this is not the case since $SO(n)$ is given as the intersection of ...
6
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2answers
94 views

A question about compact abelian groups

After learning about the duality between compact Abelian groups and discrete Abelian groups, I decided to look at exercises from various sources. One question that stood out was the following: If ...
9
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1answer
236 views

The Group of Homeomorphisms

I have been looking at Topological Groups, and I recently read about the group $\operatorname{Homeo}(X)$ of all homeomorphisms of $X$ onto itself. In particular, when $X$ is a metric space. The ...
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0answers
391 views

Haar-measure on the torus

Good evening! Let $ \mathbb{T}:=\{ z \in \mathbb{C} ; \vert z \vert =1 \} $ be the unit circle in the complex plane. We denote the trace Borel-$\sigma$-algebra on $\mathbb{T}$ by ...
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1answer
104 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 $\Gamma \cap UK = \Gamma \cap K$. ...
6
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1answer
113 views

Uniqueness of compact topology for a group

Suppose $G$ is a compact $T_2$ group. Can there be other compact $T_2$ topologies on $G$ which also turn $G$ into a topological group? ($T_2$ refers to the Hausdorff separation axiom)
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1answer
449 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
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1answer
63 views

unitary representation and denseness.

I have the next unitary representation, $\pi : G\rightarrow \mathcal{U}(H)$, where G is a closed subgroup of $S_{\infty}$ (the group of bijective functions from $\mathbb{N}\rightarrow \mathbb{N}$), ...
2
votes
1answer
122 views

When is $\: \pi_1(\langle X,\mathcal{T}\hspace{.01 in}\rangle,x_0) \:$ a topological group?

(Although I am taking an algebraic topology class, this is not homework; we have not gotten to this yet.) Let $\langle X,\mathcal{T}_X\rangle$ be a path-connected Hausdorff space. $\:$ Let $x_0$ be ...
2
votes
1answer
139 views

Which of the following topological groups are polish or locally compact?

I want to show that the next groups are polish topological groups, which criteria should I use here? And also which are locally compact (same question)? The groups are: The group of permutations ...
10
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1answer
397 views

Is a topological group action continuous if and only if all the stabilizers are open?

Let $G$ be a topological group and $(X,\mu)$ be a $G$-set, i.e. $\mu$ defines an action $X \times G \rightarrow X$. Is it then true that $\mu$ is continuous if and only if for every $x \in X$ the ...
2
votes
2answers
331 views

Locally compact topological group is Normal

How can I prove directly that a locally compact topological group G is normal? I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove ...
4
votes
1answer
246 views

Compact subgroups of the general linear group

Let $V$ be a finite-dimensional real linear space, and let $K$ be a compact subgroup of $GL(V)$ (with the usual topology); then is there a basis of $V$ such that every $f\in K$ is an orthogonal matrix ...
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1answer
86 views

Centralizers in reductive Liegroups = unimodular?

Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
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2answers
367 views

$(x,y)\to xy$ continuous but $x\to x^{-1}$ not

In the definition of topological groups we impose both $(x,y)\to xy$ and $x\to x^{-1}$ to be continuous. However, I cannot find an example where the first condition holds but the second fails. Is ...
6
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0answers
133 views

Why is any proper closed subgroup of $\mathbb{R}$ necessarily countable? [duplicate]

Possible Duplicate: Subgroup of $\mathbb{R}$ either dense or has a least positive element? If I have $G$ a closed subgroup of $\mathbb{R}$, then why is $G$ necessarily countable, except of ...
3
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1answer
160 views

$\sigma$- compact clopen subgroup.

I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact. So here's what I did so far: for $e \in U$, where $U$ is a nbhd of ...
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81 views

Induced representations of topological groups

Sorry if this is a naive question-- I'm trying to learn this stuff. If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to ...
8
votes
2answers
389 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
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2answers
184 views

For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? What about algebraic groups?

When discussing with awllower about this question, I begin to think about another one: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = ...
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1answer
347 views

Closure of the connected component of the unity is connected: is my proof valid?

I have tried to prove that a closure of a connected component of the unity in a topological group is closed, but am not sure of its validity. Since it arose from a sentence in a book on the subject,* ...
4
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1answer
332 views

Completion of Topological Group with Metric

Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...
2
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2answers
134 views

Zero Dim Topological group

I have this assertion which looks rather easy (or as always I am missing something): We have $G$ topological group which is zero dimensional, i.e it admits a basis for a topology which consists of ...
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0answers
79 views

A $T_0$ topological groups is $T_{3.5}$ (and consequently $T_3$)

I don't understand why this is so? Iv'e just seen the proof that a $T_0$ topological group is $T_1$, but don't know how to show that it's $T_{3.5}$. BTW, the fact that $x\bar{V}=\bar{xV}$, one ...
3
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1answer
85 views

String and BString

In one of the talks of J.P. May he mentioned some examples of structure groups and their classification spaces (he mentioned: O, U, SO, SU, Sp, Spin, String, Top, STop, F and SF). Most of them are ...
3
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1answer
574 views

Is the following proof valid? About the closure of a subgroup, of a topological group, being again a subgroup

I found a proof in the book Fourier Analysis On Number Fields that the closure of any subgroup is a subgroup, using the continuity argument along with the nets. Nevertheless, the following proof seems ...
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1answer
528 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
2
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1answer
45 views

question on uniform structure

It should be a triviality, I believe. The topology induced by a uniform structure $\mathcal{U}$ with $\cap \mathcal{U} =\Delta$, where $\Delta$ is the diagonal, is Hausdorff. Now I think that if I ...
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2answers
422 views

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset? I know this is true in one direction, i.e., any nonempty open subset is dense, but how ...
2
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1answer
517 views

Topological Groups

I was learning about topological groups from Atiyah-Macdonald's chapter on Completions, and I have the following question: Let $G$ be an abelian topological group. Let $H$ be the intersection of all ...
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vote
1answer
179 views

property of a topological group

This should be easy, but apparantly not for me. Let G be a topological group, and let $\mathcal{N}$ be a neighbourhood base for the identity element $e$ of $G$. Then for all $N_1,N_2 \in \mathcal{N}$, ...
9
votes
2answers
239 views

Topological rings which are manifolds

Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume ...