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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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 ...
2
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1answer
58 views
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 ...
4
votes
2answers
232 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
votes
1answer
64 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 ...
2
votes
2answers
59 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.
3
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2answers
140 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 ...
0
votes
1answer
79 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 ...
1
vote
1answer
70 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 ...
6
votes
1answer
297 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 $\mathcal{L}^1(G, \mu)$. ...
2
votes
1answer
101 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 ...
2
votes
2answers
133 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 ...
8
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3answers
225 views
How to show path-connectedness
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 ...
5
votes
1answer
162 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 ...
0
votes
1answer
77 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 ...
2
votes
2answers
361 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
votes
1answer
370 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 $G$, ...
3
votes
1answer
113 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 ...
3
votes
1answer
148 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 $G/H$ ...
7
votes
1answer
364 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 ...
6
votes
1answer
512 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 ...
5
votes
2answers
77 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 ...
8
votes
1answer
136 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
221 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 ...
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 $\Gamma \cap UK = \Gamma \cap K$. ...
6
votes
1answer
105 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)
0
votes
1answer
211 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 ...
1
vote
1answer
55 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
108 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 ...
1
vote
1answer
95 views
Criteria for a topological group to be polish.
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 ...
7
votes
1answer
190 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 ...
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vote
0answers
169 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 ...
3
votes
1answer
143 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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vote
1answer
71 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?
12
votes
2answers
345 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 ...
5
votes
0answers
122 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
votes
1answer
106 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 ...
2
votes
0answers
60 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
3answers
315 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 ...
6
votes
2answers
136 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}] = ...
1
vote
1answer
216 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
votes
1answer
264 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
votes
2answers
109 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 ...
1
vote
0answers
67 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
votes
1answer
77 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 ...
2
votes
1answer
199 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 ...
6
votes
1answer
348 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
votes
1answer
42 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 ...
1
vote
2answers
207 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 ...
1
vote
1answer
265 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 ...
1
vote
1answer
114 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 $N_1,N_2 \in \mathcal{N}$, ...
