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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325 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
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131 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 ...
7
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120 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
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86 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, ...
6
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189 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
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254 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
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301 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
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33 views

Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian. I wonder if 'homeomorphic to a torus' is enough to imply abelian. ...
4
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123 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 ...
4
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73 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
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93 views

For every space $X$, $C_p(X)$ is a topological group.

I try to show that for every space $X$, $(C_p(X), +)$ where $$+:C_p(X)\times C_p(X)\to C_p(X):(f,g)\mapsto f+g$$ and for every $x\in X$, $(f+g)(x)=f(x)+g(x)$ is a topological group. The family ...
4
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202 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
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28 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
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0answers
57 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
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87 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 ...
3
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49 views

Explicit construction of Haar measure on quotient group

Let $G$ be a locally compact Hausdorff group, $H$ a closed normal subgroup. To simplify matters we assume that the underlying topological space of $G$ has a countable base. Suppose a left Harr ...
3
votes
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98 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 ...
3
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43 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
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39 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
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74 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
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0answers
78 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
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21 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
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173 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 ...
3
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60 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
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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
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40 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
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168 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
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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
115 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
327 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
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298 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 ...
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24 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
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26 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
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36 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
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0answers
40 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
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36 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
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38 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
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35 views

There are infinitely many continuous characters on an infinite abelian topological group

Let $(G,\mathcal T)$ be an infinite abelian group and $\Bbb T$ be the circle group. Why there are infinitely many continuous homorphisms $f:G\to \Bbb T$? Is there a simple proof without using ...
2
votes
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133 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
40 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
43 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 ...
2
votes
0answers
55 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 ...
2
votes
0answers
41 views

Dense subgroup of an extremely amenble group is extremely amenable??

Recall that a topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space $K$ admit a fixed point. i.e there is a point $\xi\in K$ such that $g.\xi=\xi$ for ...
2
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48 views

A homeomorphism of topological groups.

Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If ...
2
votes
0answers
90 views

a measurable function on a LCA group coincide with an mulitplicative character almost everywhere

Let $G$ be an LCA group. We say $\tilde \chi$ is a multiplicative character if it is a continuous function : $\tilde \chi : G \to S^1 $ where $ S^1 : = \{ z \in \mathbb C : |z| =1 \}$ is the unit ...
2
votes
0answers
55 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
2
votes
0answers
46 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
2
votes
0answers
58 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
2
votes
0answers
25 views

universal nonabelian divisible group

For this post, a group $G$ shall be referred to as generally divisible, in case $\forall{x\in G:}~\forall{n\in\mathbb{N}^{\times}:}~\exists{y\in G:}~y^{n}=x$. Note. Here is no commutativity ...
2
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41 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...