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.

learn more… | top users | synonyms

20
votes
0answers
354 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 ...
9
votes
0answers
160 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
8
votes
0answers
123 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 ...
6
votes
0answers
95 views

Why is topological group not a popular topic?

In Japan, there are many universities with a formal course about topological group using the classic by Pontryagin. Yet topological group is not studied in a formal course in many other countries, ...
5
votes
0answers
306 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
5
votes
0answers
224 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
5
votes
0answers
317 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
4
votes
0answers
106 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
4
votes
0answers
121 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 ...
4
votes
0answers
80 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
4
votes
0answers
192 views

Discrete Closed Subgroup H of a Simply Connected Topological Group G isomorphic to fundamental group of G / H.

A problem in Rotman's Algebraic Topology is as follows: Given a simply connected topological group G with a closed discete normal subgroup H, show that $\pi_1(G / H) \cong H$. I believe I have this ...
4
votes
0answers
132 views

Homeomorphism groups as topological groups

As is well known, the homeomorphism group of a compact Hausdorff space is a topological group. The same is true for locally compact locally connected Hausdorff spaces, but it is false in general. Now ...
4
votes
0answers
206 views

If $G$ is a locally compact Hausdorff group, when does $G/Z$ have a probability Haar measure?

I am reading an introductory material about topological groups and the question in the tittle comes up. Due this Proposition Proposition. A locally compact Hausdorff topological group $G$ is ...
3
votes
0answers
59 views

Group actions by semi-direct products of groups

I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ...
3
votes
0answers
32 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
3
votes
0answers
62 views

Contractible pieces of $GL(n,\mathbb{C})$

Is $GL(n,\mathbb{C})$ contractible for any $n$? My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there ...
3
votes
0answers
44 views

Potential theory for LCA groups

I was wondering if there is a potential theory for locally compact abelian groups. $\textbf{Edit:}$ What are the suitable analogs for logarithmic or Newtonian potentials in the context of LCA groups ...
3
votes
0answers
231 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
3
votes
0answers
55 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 ...
3
votes
0answers
44 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 ...
3
votes
0answers
40 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 ...
3
votes
0answers
78 views

Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
3
votes
0answers
98 views

Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
3
votes
0answers
29 views

Proof of commutativity for topological I-semigroups

A topological semigroup on a closed interval I and order topology is called I-semigroup if 1 acts as an identity and 0 as an annihilator. I've seen in several articles that such I-semigroups should ...
3
votes
0answers
286 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
3
votes
0answers
66 views

A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii)) Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely ...
3
votes
0answers
36 views

The minimal divisible extension

For a prime number $p$, $F_{p}$ is the p-adic number groups and $J_{p}$ is the p-adic integer groups. Is $F_{p}$, the minimal divisible extension of $J_{p}$?
3
votes
0answers
53 views

Transitive topological action

Let $G$ be a topological group, $A$ be set and $\mu\colon G\times A\to A$ a transitive action. I'm trying to prove the statement below is false. There exists only one topology in $A$ such ...
3
votes
0answers
43 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?
3
votes
0answers
174 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 $T_1$ ...
3
votes
0answers
71 views

Induced action of topological groups

Let $G$ be a polish group, $H$ be an open subgroup of $G$ and $X$ be any metric space on which $G$ act. I want to show the following fact: If the restriction to $H $of the action of $G$ on $X$ is ...
3
votes
0answers
118 views

Group theoretic lemma about the extension of homomorphisms of profinite groups

I have a question about a group-theoretic lemma proven in Galois Groups and Fundamental Groups by Tamas Szamuely. Suppose we have a profinite group $\Gamma$, a closed normal subgroup $N \subset ...
3
votes
0answers
384 views

Lie Groups: Identity Component

Main Problem Given a Lie group. The connected component of the identity is a Lie subgroup: It is a subgroup. It is open. How to check this using topological tools? Extra Problem The quotient ...
3
votes
0answers
320 views

Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
2
votes
0answers
27 views

Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
2
votes
0answers
26 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
2
votes
0answers
80 views

Under what mild condition, inclusion of a conjugate subgroup in the initial subgroup gives equality?

Let $G$ be a group, $H \leq G$ a subgroup, and $x \in G$. If $xHx^{-1} \subset H$, then we may not have $xHx^{-1} = H$. Here is a simple counterexample. But this don't prevent the hope to finding ...
2
votes
0answers
35 views

The path-connected field is just $\mathbb{R}$ or $\mathbb{C}$?

A topological ring is a ring $R$ which is also a topological space such that both the addition and the multiplication are continuous as maps. $F$ is a topological field, if $F$ is a topological ring, ...
2
votes
0answers
37 views

A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
2
votes
0answers
43 views

Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is ...
2
votes
0answers
30 views

Smooth admissible representations, Hom, tensor and extension of scalars.

Let $G$ be a locally profinite group, and consider $V$ and $W$ smooth admissible representations of $G$ over some field $F$ (or char. $0$). Let $E/F$ be any field extension. I'd like to find ...
2
votes
0answers
28 views

Amenability; topology on power sets

I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem ...
2
votes
0answers
30 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
2
votes
0answers
38 views

suspension foliations on thickened surfaces

I've seen this statement without proof in a peer reviewed journal and I'm looking for a proof: "If $L$ is an oriented surface with boundary($\neq D^2$), and $C$ is a designated boundary component, ...
2
votes
0answers
41 views

Quotient group of $\mathbb{R}^2$ by irrational line

In a section about topological groups, exercise 4.10 in I.M. James' General Topology and Homotopy Theory asks Show that for irrational values of $\alpha$ the factor group of the real plane ...
2
votes
0answers
42 views

When can a group be made into a ring? How `little' of the ring structure must be specified?

Given a (topological) abelian group $G$ and a (bicontinuous) $G$-bilinear map $\mu: G \times G \to G$, clearly $G$ becomes a (topological) ring by specifying $$ x y := \mu(x, y) \quad \forall x, y \in ...
2
votes
0answers
45 views

Sufficient to check intersection of sub base elements with a dense set in compact open topology

I am reading a book about duality and there was this following claim: If I have a compact group G* (dual group of G, and G is discrete) with the compact open topology, then for any A, a subset of G* ...
2
votes
0answers
154 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
2
votes
0answers
44 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 ...
2
votes
0answers
72 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 ...