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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How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ? I would really really appreciate if I can get a step by step ...
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48 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
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35 views

Proof that a particular subgroup is proper

I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me. Context: I have a pro-$p$ group $G$, which just means ...
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27 views

In the quotient $G/H$, why we must suppose that $H$ is closed?

We have the following known statement: Theorem: If $G$ is a topological group and $H\subseteq G$ is a closed invariant subgroup of $G$, then $G/H$ (of course with the quotient topology) is a ...
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35 views

Question about Modular function in Haar measure

I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that: Let $G$ be a locally ...
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174 views

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
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75 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
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9 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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28 views

Isomorphisms of LCA Groups

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
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26 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
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55 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
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27 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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1answer
20 views

Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
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16 views

A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
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1answer
46 views

The natural isomorphism in the Pontryagin Duality

Pontryagin duality is the statement that there's a natural isomorphism between the identity functor on $\mathsf{LCA}$, the category of locally compact (Hausdorff) abelian groups with continuous ...
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1answer
51 views

Prove that $SL(n,R)$ is connected.

Prove that $SL(n,R)$ is connected. The problem is I know only topological groups from Munkres only. Again Just started fundamental groups. So if anyone can explain me how it is true in a lucid ...
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64 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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1answer
56 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
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46 views

Show that every topological group is $T_3$

I know that it is sufficient to show that for one point ($e$) and any neighbourhood $U$ of $e$, we have a neighbouhood $V$ with $\bar{V} \subseteq U$. Since $x \to x^{-1}$ is continuous, it follows ...
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32 views

Cech-complete separable groups

It is well known that a Baire measurable homomorphism between Polish groups is continuous. Is the same true if we replace Polish groups by Cech-complete, separable groups? It is not true ...
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21 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
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37 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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56 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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41 views

question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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265 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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50 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
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67 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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68 views

if $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?

Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) If $\pi_1(G)$ is ...
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23 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
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42 views

Translation invariant metrics and topological groups

During a lecture was pointed out that one of the main feature, from a topological perspective, of normed vector spaces is the translational invariance, that is that one can study the topological ...
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43 views

Reference for Topological Groups

Topological groups were a topic that were covered minimally at my undergraduate institution but it's a topic that I'm finding a need quite a bit in the number theory I'm reading (class field theory). ...
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1answer
36 views

Existence of an open normal subgroup of a neighborhood of 1 in a compact Hausdorff and totally disconnected topological group

Let $G$ a compact Hausdorff and totally disconnected topological group. Then every neighborhood of 1 contains an open normal subgroup of finite index in $G$. I need this lemma to prove that every ...
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23 views

Component of the identity is generated by any connected neighborhood of the identity when the group is locally connected?

I read a theorem that if $G$ is a locally connected group, then the component of the identity $G_0$ is generated by any connected neighborhood of $e$. It goes like: Let $V$ be a connected nbhd of ...
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1answer
67 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
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1answer
44 views

Non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability

I think non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability are pain in the ass. The major trouble is that the Haar measures on them are not necessarily ...
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1answer
54 views

Inducing a surface area measure on $S^2$ from the Haar measure on $SO(3)$

I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example ...
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44 views

Can topological groups be smoothed into lie groups?

I've been thinking about this for the past couple of days and I'm really not sure of the answer.. By "smoothed" I mean that for any arbitrary precision we can find a Lie group which approximates the ...
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288 views

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
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1answer
19 views

Soft: Interpretation of a periodic event on circle group

Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group ...
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1answer
37 views

Loop spaces have the homotopy type of a topological groups

Every based loop space has the homotopy type of a topological group. I would like to understand this fact, and this is what this question is about : why is it true, and how does one prove it? I ...
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Putting a direct system on a product of direct limits

I was going back through some class notes discussing the direct limit topology (final topology) and we showed that the direct limit of topological groups is a topological group. To do this, we showed ...
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33 views

What is the restricted product categorically?

The restricted product is a construction for locally compact abelian topological groups. Let $I$ be an indexing set, with $J$ some finite subset. Let $G_i$ be a locally compact topological group ...
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22 views

Homomorphisms on a solenoid

Let $G$ be an additive subgroup of $\mathbb{Q}$ containing $\mathbb{Z}$ i.e., $\mathbb{Z} < G < \mathbb{Q}$. A homomorphism on $G$ is of the form $\phi_r : G \rightarrow G$ given by ...
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Can a countable group have uncountably many distinct Hausdorff group topologies?

Question. Can a countable group have an uncountable number of distinct Hausdorff group topologies? By a group topology one understands a topology with respect to which the group operations are ...
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82 views

Is a partially topological group completely regular

Let $G$ be a group and $\mathcal T$ be a topology on $G$ and the function $$ \begin{align*} &f:G\times G\to G\\ &f(x,y)=xy^{-1} \end{align*} $$ be continuous at $(1,1)$. Is $(G,\mathcal T)$ ...
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When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?

Suppose you have a group $G$ acting on $ (M,d)$ a compact metric space by isometries (meaning $d(gx,gy) = d(x,y)$ for all $x,y \in M$ and all $g \in G$), transitively and faithfully. You can define ...
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35 views

Is there a natural (non-trivial) topology for the automorphism group of a locally compact abelian group?

I've been thinking about automorphism groups of locally compact abelian groups somewhat over the last few days. It's not hard to see that in the case of the torus and integers, the (continuous) ...
3
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2answers
112 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
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17 views

Is this a compact group?

Consider $$x(t)=e^{-iHt}x(0)$$ and define $$G=\{ e^{-iHt}\mid t\in \mathbb{R}_{\geq 0}\}$$ Also write $\bar{G}$ to the closure of $G$ wrt the Euclidean topology. Q: is $\bar{G}$ a compact group? ...
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1answer
99 views

What is a Topological Group Intuitively?

What is a topological group intuitively, beyond just being able to say things are near to each other in a group, and why is it a good idea to consider this theory as part of general topology as ...