Questions tagged [topological-groups]
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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GNS representations with $\hat{G}$ and $C^*$-algebras
Let $G$ be a commutative discrete group and $\mathcal{A} := l^1(G)$. Given a linear positive functional $\mu$ on $\mathcal{A}$, consider the GNS $(\mathcal{H}_\mu, \pi_\mu, \xi_\mu)$. Define $\mathcal{...
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G abelian Lie group, $(\pi, V)$ a finite-dim unitary rep. $\exists $ mutually orthog. 1-dim invariant linear subspaces s.t $V= \bigoplus_i V_i$
Let G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G.
I proved that
$\pi \text{ is irreducible iff dim } V = 1\tag 1$
Using this I want to show that
There exist ...
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details in the proof: G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G. Then $\pi$ is irreducible iff dim $V = 1$
I am getting lost at filling in the details in the proof that
Let G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G. Then
$\pi$ is irreducible iff dim $V = 1$
I having ...
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Finding open subgroups of profinite groups from description as an inverse limit
For the profinite group $\widehat{\mathbb Z} := \varprojlim_n \mathbb Z/n\mathbb Z$ (which ofcourse actually has a ring structure as well), you can show its open subgroups are precisely the groups $n \...
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$(2)$ is not open in the $(6)$-adic topology
Suppose we have the ring $\mathbb{Z}$ and the ideal $I=(6)$ on it. Thus we can talk about the $(6)$-adic topology. Supposedly, $\mathbb{Z}$ with the $(6)$-adic topology is a topological ring. For that,...
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Accumulation point in a topological group of orthogonal matrices over R
I would like to ask your opinion on the point that looks simple.
Consider the group of orthogonal matrices of order n over the field R of reals, equipped with the topology induced by the Euclidean ...
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Why is coboundary $\sigma \to \sigma m-m$ automatically continuous?
Let $K$ be a field. Let $G_K := Gal(\overline{K}/K)$ be absolute Galois group of $K$.
Let $M$ be a $G_K$-module, that is, $G_K$ (with the Krull topology) acts continuously on $M$ with discrete ...
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Prove that the closure $\overline{H}$ of a subgroup $H \subseteq G$ of a topological group, is a subgroup. [duplicate]
I'm tackling the problems in `Introduction to Topological Manifolds' by Lee, and this is the first time I've been tackling with topological groups. This is problem 13-9:
"Let $G$ be a topological ...
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Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.
If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
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Finite covers of $X$ and continuous profinite group actions
I have a question that is related to the one here, but I found the answer there unsatisfactory and kind of confusing.
For context, we are considering the action of the fundamental group $\pi_1(X,x)$ ...
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showing that this specific functional is positive on $l^1(G)$ where $G$ is a discrete topological commutative group
Let $G$ be a discrete commutative group, and $\phi:G \rightarrow \mathbb{T}$ be a group homomorphism where $\mathbb{T}$ is the circle i.e. $\mathbb{T} := \{z \in \mathbb{Z}:|z| = 1\}$. Consider $l^1(G)...
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Is an Archimedean topological group of the reals isomorphic to $(\mathbb R,+)$?
A group $H:=(\mathbb R,\boxplus)$, is given to be
Ordered as per the canonical order of $\mathbb R$.
Archimedean as per order in 1.
Topological as per canonical topology of $\mathbb R$.
Can it be ...
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Measure on the set of orbits of a flow via the von Neumann crossed products?
Suppose we're given an action (possibly: ergodic) of a group G (possibly: uncountable) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question/reference request: is there a ...
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Maps from a finitely generated pro-p group to $\mathbb F_p$ factors through Frattini quotient
Let $G$ be a finitely generated pro-p group, these notes (p.99, Corollary 5.4.21) claims that all maps from $G$ to $\mathbb F_p$ factor through the Frattini quotient ($G/\Phi(G)$), where $\Phi(G)$ is ...
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Topological definition of Spin$(p,q)$?
In short, How can we define Spin(p, q) without referencing Clifford algebras? The answer should be something like "Spin$(p, q)$ is the unique double cover of SO$^+(p, q)$ such that ...". ...
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A profinite group is the inverse limit of all quotients by open normal subgroups
My definition of a profinite group is it's the inverse limit of a system of finite groups, $G=\varprojlim_{i\in I} G_i$.
But I can't see why $G=\varprojlim_{j\in J} G/N_j$, where $\{N_j\}_{j\in J}$ is ...
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Normal closure of closed subgroup is closed in a f.g. profinite group?
Is there an example of a topologically finitely generated profinite group $G$ and a closed subgroup $H$ such that we simultaneously have:
The profinite normal closure $L$ of $H$ is open in $G$.
The ...
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How is this group action over a colimit continuous?
In their book "Sheaves in Geometry and Logic" (paragraph III.9: continuous group action) Saunders MacLane and Ieke Moerdijk define a complete subcategory $\mathsf{S}(G)$ of a category $\...
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Is there a manifold $M$ with nontrivial $\pi_2(M)$ but trivial $\pi_3(M)$?
Forgive me if the following questions on homotopy groups are trivial - I am not knowledgeable in the subject.
Is there a manifold $M$ with nontrivial second homotopy group $\pi_2(M)$ but trivial third ...
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Proving that $G/H$ is Hausdorff [closed]
Let $G$ be a topological group and $H$ be a normal subgroup. I want to show that $G/H$ is a topological group. I have managed to prove that $m:G/H\times G/H\to G/H, \bar a \bar b\to \bar{ab}$ and $i:G/...
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Is it true that every residually finite group is a profinite group under some topology?
I know that a profinite group is residually finite. I am interested in the converse, which seems true. My reasoning is as follows. Please let me know if I am making some mistakes.
Let $G$ be a ...
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Definition of profinite group
Consider the following definition:
A profinite group is a topological group that is compact, Hausdorff, and admits a neighborhood basis of $1$ made of normal subgroups.
My question is: it is ...
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Artin's theorem in the infinite dimensional case
I am talking about the theorem from Galois theory, specifically to Milne's course notes on field theory proposition 7.10 (which can be easily found online).
In the proof of this proposition, we have ...
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If $A : G \times M \rightarrow M$ is a proper action and that $M$ is a metric space, the quotient space $M/G$ is Hausdorff
Def 1: Let $f : X \rightarrow Y$ be a continuous map. $X,Y$ Topological spaces. $f$ is called proper if $f^{-1}(K)$ is compact for every compact $K \subseteq Y$.
Def 2: G : topological group. A ...
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A continuous action of a compact topological group on a Hausdorff space is proper [duplicate]
Definition: G : topological group. A continuous action $\alpha : G \times M \rightarrow M$ is
called a proper action if the following map is proper:
$(\alpha; \text{id}) : G \times M \rightarrow M \...
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Problem 5-a in Dynamics in one complex variable book
Given a sequence of numbers $1 > a_1 > a_2 > \cdots$ converging to $0$, let $U \subset \mathbb{C}$ be obtained from the open unit square $(0,1)\times(0,1)$ by removing the line $[a_n, 1]\...
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$\overline Y=\varprojlim \pi_i(Y)$ [closed]
Everything is in the title !
Ribes & Zaleski propose (cor 1.1.8 of"Profinite Groups") as an exercise to prove $\overline Y=\varprojlim \pi_i(Y)$ where $Y\subset\varprojlim E_i$ is a ...
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Are the left uniformly continuous functions on a topological group dense in $L^1(Haar)$?
Let $G$ be a locally compact second countable topological group (Lie group if it helps). A function $f : G \to \mathbb{C}$ is left uniformly continuous if for all $\epsilon > 0$ there exists an ...
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Using Munkres formulation of the Hopf Trace formula to finish a proof of the Lefschetz Fixed Point Theorem
I am very inexperienced with Algebraic Topology but I am trying to put together a proof for the Lefschetz Fixed Point Theorem, that I can understand, for a presentation in class.
I am following ...
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Topological group: if $g_n, h_n, k_n \in G$ s.t. $g_n h_n \to e$ and $k_n \to e$ does $g_n k_n h_n \to e$?
This is certainly true if $g_n$ (equivalently $h_n$) converge in $G$ and it feels like it should be true but after trying to prove it, it's unclear.
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Proof about certain family of sets that is a basis for the open-compact topology of homomorphism from a topological group to the unit circle
Let $G$ be a locally compact topological group and $\hat{G}$ the set of homomorphisms from $G$ to the unit circle $\mathbb{S}^1 = \{ z \in \mathbb{C} : |z| =1 \}$. For every compact set $K \subseteq G$...
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Is there any canonical way to find out the Lie group corresponding to a manifold?
Lie groups are manifolds. But it is well known that not all manifolds are Lie groups. E.g. There are no Lie groups that correspond to the 2-sphere. More generally, the fundamental group has to be a ...
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Countable Connected Hausdorff groups
Does there exist a countable connected Hausdorff (nontrivial) topological group?
I'm aware that countable connected Hausdorff spaces exist (for example the Bing space, the Golomb space, or $\mathbb{Q}...
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Characterization of the completion of a first-countable topological abelian group
$\def\codom{\operatorname{codom}}$If $G$ is a first-countable abelian topological group, one can find a morphism $G\to\hat{G}$ of abelian topological groups, with $\hat{G}$ first-countable and ...
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Show that some quotient space of $GL_2(\mathbb{Q}_p) $ is compact
Let $T_{\tau}:=\{\begin{pmatrix}a&b\tau\\b&a\end{pmatrix}:a,b\in\mathbb{Q}_p,a^2-b^2\tau\neq0\}$, where $\tau\in\mathbb{Q}_p^{\star}$ with $\tau\notin(\mathbb{Q}_p^{\star})^2$.
Author claimed ...
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Consequences of differing definitions of topological groups
I'm using this document to learn about the representation theory of compact groups. They define a topological group $G$ to be a group with a topology such that
The multiplication and inversion ...
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Pontlyagin dual of direct sum,$ \widehat{\bigoplus_{i\in \Lambda} M_i} = \prod_{i\in \Lambda} \hat{M_i}. $
Let $M_i$ (for $i \in \Lambda$) be a family of abelian groups.
Let $\bigoplus_{i\in \Lambda} M_i$ denote the infinite direct sum of the groups $M_i$.
Let $\hat{M}$ denote the Pontryagin dual of the ...
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Linear topology
Definition: A linear topology $\tau$ on a left $A$-module $M$ is a topology on $M$ that is invariant under translations and admits a fundamental system of neighborhood of $0$ that consists of ...
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Doubts in the proof that a subgroup $H$ of a Lie group $G$ that is a submanifold of $G$ is (topologically) closed in $G$
I am trying to understand the details in the proof of
Let $G$ be a Lie group, $H $ a subgroup. then
$H$ is a submanifold of $G \implies H$ is closed in $G$
proof:
Since $H$ is a submanifold of $G$, ...
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Every continuous action $\varphi$ on $S^1\cup \{(x, 0): x\in[-1, 1]\}$ does have $(-1, 0)$ as fixed point
If $X=S^1\cup \{(x, 0): x\in[-1, 1]\}$, then for every homeomorphism $F:X\to X$, $F((-1, 0))=(-1, 0)$. This implies that if $\varphi:G\times X\to X$ is a continuous action, then $\varphi(g, (-1, 0))= (...
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Embedding of topological group in its group of self-homeomorphisms
The second part of the answer to the question
When does a topological group embed topologically in its group of homeomorphisms?
made me wary about an argument I have which seems easier and more ...
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Circle with infinite corners
A vertex is defined as a 'meeting point of two lines that form an angle'.
When I increment the circumference of a circle into infinitesimal small
increments I get something like shown below in the ...
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Quotient by compact topological group has closed projection
Let $X$ a topological space and $G$ a compact topological group, why is the quotient map
$$\pi \colon X \to X/G$$
closed?
For every closed subset $C$ of $X$
$$\pi^{-1}(\pi(C)) = \bigcup_{g \in G}g\...
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Equivariant Cohomologies of Two Homotopy Equivalent Topological Groups
Suppose that $K \subset G$ is a topological subgroup and this inclusion is a homotopy equivalence (so somewhat stronger than what's written in the title). I'm not assuming compactness but am happy to ...
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Additional sufficient conditions for closed connected Lie subgroups
The textbook counterexamples of connected Lie subgroups that are not closed seem to all be constructed by having one-parameter subgroups not being closed, e.g. the irrational winding of the Torus. I'...
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The closure of $\{0\}$ in $(Z, +)$ endowed with the p-adic topology
Let $p$ be a prime and $(Z, +)$ the additive group of intergers. Consider the filter
$$V_p = \{U \subset \mathbb{Z} \mid \ \text{exists} \ n \in \mathbb{N} \ \text{such that}\ p^n\mathbb{Z} \subset ...
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Universal property of the completion of a first-countable abelian topological group
Let $G$ be a first-countable abelian topological group. Its completion $\hat{G}$, set-theoretically, is the family of Cauchy sequences in $G$ modulo the equivalence relation $(x_n)\sim (y_n)$ iff $x_n-...
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When can monothetic groups be turned into rings?
If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image.
A multiplication can be naturally defined ...
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Is this group isomorphic to the real numbers?
I have a locally compact abelian group $G$ with the following properties:
It is connected (therefore divisible) and non-compact;
It admits a $\mathbb{Q}$-vector space structure and for any $g \neq 0$,...
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Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
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