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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Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$

I would like to know topologically what the space $(\mathbb{C}^n - \{0\})/\mathbb{Z}_k$ may be thought of as. The paper I am reading says that we let $\mathbb{Z}_k$ act on $\mathbb{C}^n - \{0\}$ via ...
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44 views

Discrete VS finite groups

I am having some troubles to understand what the difference is between discrete and finite groups. I know they are defined differently, I can't quite understand the difference. I am guessing every ...
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2answers
24 views

Is there such thing as an $n$-fold cover of $SO(n)$?

I know already that the $Spin(n)$ group is a double cover of $SO(n)$, but is there such a thing as a triple cover, fourfold or $n$-fold cover? I am interested to know if there are 3-spinors (?) or ...
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63 views

Real orthogonal matrices with fixed entry value has zero Haar-measure?

Consider the set of real orthogonal matrices of size $n \times n$ such that one entry, say $a_{i,j}$ for fixed $i$ and $j$, satisfies $a_{i,j}=0$. Has this set zero Haar-measure? A simple proof, in ...
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34 views

Classification of symmetric space of non compact type

Is there a classification of rank one symmetric space of non compact type ? Remark, for rank 1, There are three families : 1) hyperbolic n-space, corresponding to the Lie group SO(n,1). 2) complex ...
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31 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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53 views

Absolute Galois group and continuous homomorphism.

Let $G=\mathrm{Gal}(\bar K/K)$ be the absolute Galois group of a number field $K$. I have some problem to understand that the homomoprhism $$ \rho :G \longrightarrow \mathrm{GL}_r(E)$$ $E$ a finite ...
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20 views

What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...
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53 views

Examples of left-topological compact semigroups

I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$. Several results about left-topological compact ...
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38 views

If a group and a groupoid have the same profinite completions, can anything be said about them?

Pretty much what I ask in the title. Suppose $\mathbb{G}$ is a group and $\mathcal{G}$ is a groupoid. Consider their profinite completions $\overline{\mathbb{G}}$ and $\overline{\mathcal{G}}$. If ...
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31 views

Is there an relation between the notions of continuous and discontinuous group actions?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an ...
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46 views

Continuity using topological groups

I have been give this question. I am just a bit confused where to start? Prove that $g : \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\to \mathrm{GL}(n,\mathbb R)$ given by $g(x, y) := xy$ ...
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1answer
18 views

Compact subgroups of a topological group

Consider $G=(0,\infty )$ , with the metric induced by Euclidean metric from $\mathbb R$ .$G$ is a group under multiplication . Then which subgroups of it are compact $?$ Now ...
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32 views

What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?

I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices. Is there an intuitive way to think about it, ...
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28 views

Neighborhood base at the unit element in a topological group

One can define the open set $U$ be such that $U$ is open if for every $x\in U$, $xV\subset U$ for some $V\in\mathcal{N}$. (2) and (3) would give the desired continuity for group multiplication and ...
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28 views

Proving the Existence of n-linked knots

I was reading up on knots and links and came across: The Hopf Link: https://en.wikipedia.org/wiki/Hopf_link Solomon's Knot (Double Link): https://en.wikipedia.org/wiki/Solomon%27s_knot Which got me ...
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1answer
42 views

Topological groups, is continuity a requirement or consequence of this definition?

On the Wikipedia-page for topological groups, it is stated as a requirement that the binary operation and the inverse operation is continuous. In Munkres he makes this definition: A topological ...
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27 views

Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on ...
3
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1answer
61 views

Fubini's theorem for topological monoids

A topological commutative monoid is a commutative monoid $(X,+)$ with a topology $\cal O$ such that $+$ is a continuous function. There is a notion of infinite sum on topological commutative monoids, ...
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31 views

How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
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131 views

Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)
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1answer
45 views

Classifying topologically cyclic groups

The group of p-adic integers has a dense cyclic subgroup, i.e. it is topologically cyclic What are some (or all) other non-trivial (i.e. non-discrete) examples of such groups?
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93 views

Topological group, a big exercise!

Let $(G,d)$ be a group that is also a compact metric space, and G is topological (The maps $G \times G \to G, (x,y) \to xy$ and $G \to G, x \to x^{-1}$ are continuous). And then we consider ...
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40 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 ...
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139 views

Can there be a Diaconis-Shahshahani Upper Bound Lemma for Compact Groups?

Let $G$ be a finite group and $\nu\in M_p(G)\subset \mathbb{C} G$ a probability measure on $G$ and let $\pi$ be the uniform distribution on $G$. Denote by $d_\rho$ the dimension of a non-trivial ...
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25 views

Is $(X_G, d_G)$ a compact manifold?

Let $G$ a compact topological group act on $(X,d)$ by isometries. We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It ...
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1answer
48 views

Definition of Roelcke uniformity

Let $(G,\mathcal T)$ be a topological group. The set of all uniformities on $G$ forms a lattice $\frak A$ and the set of all uniformities on $G$ producing $\cal T$, forms a lattice $\frak B$. The ...
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1answer
105 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
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31 views

Explicit construction of Haar measure on a locally profinite group

Let $G$ be a locally profinite group. A Haar measure $\mu$ on $G$ is a measure defined on the $\sigma$-algebra $\mathcal B(G)$ of all Borel sets of $G$ with the following properties. 1) $\mu(K) ...
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36 views

Two disjoint compact sets in a topological group

Let $(G, \cdot )$ be a compact (Hausdorff) topological group. If $A$ and $B$ are two disjoint compact subsets of $G$, how can we show that there exists a nonempty open set $V$ such $A\cdot ...
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115 views

Explicit construction of Haar measure on a profinite group

Let $G$ be a profinite group. It is known that in $G$, every neighborhood of the identity element contains an open compact subgroup. I would like to explicitly construct the Haar measure on $G$. The ...
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1answer
77 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
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39 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 ...
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1answer
102 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
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1answer
23 views

equality in uniform space and topological groups

I wanted to ask the following: If I have a topological group $G$, I know I can create a base for a uniform space as follows: for each $U$ a neighborhood of e, we define $V_u= \{(x,y):x^{-1}y\in U\}$. ...
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1answer
57 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
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52 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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30 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 ...
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1answer
59 views

Symmetric open neighborhood in topological group

I want to prove that ($\ast$) If $G$ is a Hausdorff topological group, then for every neighbourhood $U$ of $e$, there exists a symmetric neighbourhood $V$ of $e$ ($V^{-1}=V$), s.t $V*V \subset U$ ...
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1answer
45 views

Quotient group and kernel of canonical projection

Imagine we have a group $G$ acting properly and freely (as a group action $\Phi: G \times M \rightarrow M$) on a manifold $M$, then $M/G$ is a manifold and there is a smooth submersion $\pi: M ...
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56 views

Who are the mathematicians in US who are working on expander graphs right now?

I am familiar with only the "big" names doing this research like Gharan, Nikhil Srivastava, Dan Spielman, Jean Bourgain, Luca Trevisan, Elina Fuchs, Peter Sarnak , Amin Saberi and Terence Tao. I would ...
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1answer
41 views

Topology open balls and open sets

Let $A\subseteq \mathbb{R}^n$ be an open set Prove that A can be written as a countable union of open balls. We thought about it having to do with creating a ball with a radius of $r$ ...
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1answer
76 views

Show that $\widehat{\mathbb Z}\cong \prod_{p\;\text{prime number}}\mathbb Z_p$

Let $\widehat{\mathbb Z}=\varprojlim_n \; \mathbb Z/n\mathbb Z$ be the inverse limit of the inverse system $(\mathbb Z/n\mathbb Z)_{n\in \mathbb N}$ and let $\mathbb Z_p=\;\varprojlim_n\; \mathbb ...
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1answer
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Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
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39 views

Every representation of compact group is a direct sum of irreducible

Recently I asked about (references to) some results concerning representation theory of compact topological groups: here is the discussion Representation theory of locally compact groups In ...
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16 views

Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
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2answers
138 views

Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
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1answer
22 views

Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen

I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...
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1answer
53 views

$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$

Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times ...
3
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68 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 ...