1
vote
1answer
19 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 ...
2
votes
2answers
109 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 ...
3
votes
2answers
62 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
2
votes
0answers
36 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. ...
0
votes
0answers
46 views

looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
2
votes
3answers
148 views

What is an awesome book as an introduction to hyper groups

I'm a grad studen and i'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that i need to ...
2
votes
1answer
25 views

Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
4
votes
0answers
42 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 ...
2
votes
1answer
46 views

Familiarizing with the Grothendieck topos $\mathbf{B}G$.

I am trying to familiarize with the Grothendieck topos $\mathbf{B}G$ of continuous $G$-sets, where $G$ is a topological group. I am unfortunately not very familiar with working with different ...
3
votes
1answer
67 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
10
votes
1answer
115 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
4
votes
1answer
59 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
1
vote
0answers
46 views

The structure of locally compact abelian groups by D. L. Armacost.

I do not have access to this important book: The structure of locally compact abelian groups by D. L. Armacost. I need to the Example 6.4 of this book. I will be grateful if you can help
11
votes
1answer
160 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact ...
3
votes
1answer
70 views

Representation of topological groups

I am looking for a good book of topological representation. I have a very good insight of representation theory of finite groups, and I want to explore topological representations. I saw a book by ...
4
votes
1answer
194 views

A short exact sequence of groups and their classifying spaces

Suppose that we have a short exact sequence of topological groups: $$1 \to H \to G \to K \to 1.$$ I have found some papers mentioning that the above sequence induces a fibration: $$BH \to BG \to BK.$$ ...
7
votes
2answers
272 views

Good book on topological group theory?

I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
1
vote
0answers
60 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
1
vote
0answers
36 views

History: continuously differentiable groups over the real numbers

Continuously differentiable groups over the real numbers are all isomorphic to addition, as is well-known, but who proved it and when?
3
votes
1answer
164 views

Research Sources for $SL(2,R)$

Can anyone guide me to a good site for the special linear group $SL(2,R)$, especially one that goes deep into its subgroup and normal subgroup? Book recommendations would be great too.
11
votes
1answer
199 views

subgroup of connected locally compact group

I need a reference or a short proof for the following property: A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
0
votes
1answer
90 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
83 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?
8
votes
2answers
376 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 ...
4
votes
1answer
319 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 ...