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1 answer
342 views

Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$

The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
Dominic van der Zypen's user avatar
0 votes
1 answer
98 views

Is every subgroup closed in this complete, nondiscrete topological group?

Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
Nick Belane's user avatar
4 votes
0 answers
97 views

Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?

Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $? The assumption that $ N_G(H)/H $ is finite cannot be weakened ...
Ian Gershon Teixeira's user avatar
5 votes
0 answers
96 views

$M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?

Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
YC Su's user avatar
  • 605
5 votes
1 answer
251 views

In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
Linus's user avatar
  • 658
0 votes
0 answers
51 views

Approximating open subset of profinite group by union of cosets of ideal

I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
jb1403's user avatar
  • 1
2 votes
0 answers
164 views

Triviality of map $(\Sigma \theta)^*$

We know that there is a cofibration sequence $$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
Sajjad Mohammadi's user avatar
4 votes
2 answers
292 views

$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$ Is this fiber sequence induced from a short exact ...
zeta's user avatar
  • 447
5 votes
0 answers
249 views

Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
Chicken feed's user avatar
0 votes
0 answers
70 views

A cellular automaton with an image that is not closed

Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
mahdi meisami's user avatar
8 votes
1 answer
181 views

Stone-topological/profinite equivalence for quandles

A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$: (Q1) ...
Alex Byard's user avatar
0 votes
0 answers
96 views

Idempotent conjecture and (weak) connectivity of (a reasonable) dual group

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of ...
Ali Taghavi's user avatar
4 votes
1 answer
233 views

Profinite groups with isomorphic proper, dense subgroups are isomorphic

I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
Alex Byard's user avatar
4 votes
0 answers
425 views

Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$

Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3). How to show the composition $$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$ is non-trivial ...
Sajjad Mohammadi's user avatar
5 votes
1 answer
287 views

Extreme amenability of topological groups and invariant means

Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
Muduri's user avatar
  • 225
0 votes
0 answers
123 views

Classification of closures of additive subgroups of $\mathbb{R}^n$

If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either $\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or $\...
Nate Ackerman's user avatar
4 votes
1 answer
223 views

Existence of disintegrations for improper priors on locally-compact groups

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
Tom LaGatta's user avatar
  • 8,512
4 votes
0 answers
74 views

Is each TS-topologizable group TG-topologizable?

Definition 1. A topology $\tau$ on a group $X$ is called $\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$; $\bullet$ a ...
Taras Banakh's user avatar
  • 41.8k
7 votes
0 answers
138 views

The smallest cardinality of a cover of a group by algebraic sets

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
Taras Banakh's user avatar
  • 41.8k
0 votes
0 answers
163 views

Presentation complex of a finite perfect group and its features

Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions: Is there any special property of $X_G$ due to the group's perfectness? What can we say ...
piper1967's user avatar
  • 1,177
6 votes
1 answer
535 views

Finite *covering* groups that act freely on some sphere

A remarkable result (reviewed here) -- going back, at least, to P. A. Smith, developed by Cartan & Eilenberg and Milnor, and culminating in the theorem of Madsen, Thomas & Wall -- ...
Urs Schreiber's user avatar
2 votes
0 answers
222 views

Complete reducibility, in linear algebra and in topology

I thought that this is a simple question and asked it at the Mathematics StackExchange, but I now have to elevate it to MathOverflow. Consider a representation $A(G)$ of a group $G$ in a vector space $...
Michael_1812's user avatar
1 vote
1 answer
235 views

Group structure on the strip

Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits_{l\in \{a,b\}}\{d(x,l)\}$. Let $(X,*)$ be a topological group with the following ...
Ben Tom's user avatar
  • 107
9 votes
1 answer
410 views

On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces

In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
Alessandro Codenotti's user avatar
4 votes
2 answers
382 views

Topology on the hom space between profinite groups

$\DeclareMathOperator\Hom{Hom}$Let $G,H$ be profinite groups. Let $\Hom(G,H)$ be the set of continuous group homomorphisms, equipped with the compact-open topology. I'd like to understand the ...
stupid_question_bot's user avatar
18 votes
0 answers
1k views

What is the strongest nerve lemma?

The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology: If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
2xThink's user avatar
  • 81
3 votes
0 answers
282 views

Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra $L_0 \to L_1 \to L_2 \to \cdots$ obtained as follows. Take $L_0$ as some $S_g$, an ...
Shijie Gu's user avatar
  • 2,083
2 votes
1 answer
217 views

A variation of closed-subgroup theorem

$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group. I am pretty sure that this theorem should have a "...
aglearner's user avatar
  • 14.3k
-8 votes
1 answer
351 views

Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
firn's user avatar
  • 23
4 votes
0 answers
72 views

When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
Nassim's user avatar
  • 51
0 votes
0 answers
152 views

Left-side cosets of an open subgroup

Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
M masa's user avatar
  • 479
8 votes
1 answer
509 views

About locally compact groups without compact subgroups

Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
Wlod AA's user avatar
  • 4,786
9 votes
2 answers
902 views

Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?

Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I ...
Adam's user avatar
  • 2,390
9 votes
2 answers
505 views

A natural $\mathbb Q\times \mathbb P$ subset of $\mathbb R$?

I would like a simple description of a dense subset of $\mathbb R$ which is homeomorphic to $\mathbb Q\times \mathbb P$. Preferably the description will be of an algebraic nature, and perhaps the set ...
D.S. Lipham's user avatar
  • 3,317
3 votes
2 answers
300 views

Upper density of subsets of an amenable group

Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper ...
Otto's user avatar
  • 1,006
14 votes
2 answers
502 views

Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...
YCor's user avatar
  • 63.9k
16 votes
1 answer
502 views

Group actions and "transfinite dynamics"

$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
Burak's user avatar
  • 4,265
15 votes
1 answer
512 views

fundamental groups of complements to countable subsets of the plane

This question is a follow-up of this MSE post and a comment by Henno Brandsma: Question 1. Let $S$ be the set of isomorphism classes of fundamental groups $\pi_1(E^2 - C)$, where $C$ ranges over all ...
Moishe Kohan's user avatar
  • 12.3k
4 votes
1 answer
276 views

Shifting the group homology of a topological group?

Let $G$ be a topological group. It has a classifying space $BG$, which has homology groups $H_{*}BG$. Changing the topology of $G$ affects the space $BG$ and hence its homology groups. For example ...
John Greenwood's user avatar
11 votes
1 answer
991 views

Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
D.S. Lipham's user avatar
  • 3,317
-2 votes
1 answer
131 views

$G$- space is locally compact [closed]

Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
math112358's user avatar
2 votes
1 answer
82 views

Structure of extensions arising in Lie approximation of connected groups

My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known: Let $G$ be a connected, locally compact, Hausdorff group, ...
Yemon Choi's user avatar
  • 25.8k
5 votes
0 answers
316 views

Polish groups with no small subgroups

Definitions. A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space. A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
Jackson Morrow's user avatar
5 votes
2 answers
328 views

Set of topologies on a group making it a compact Hausdorff topological group

Maybe stupid, but from the following well known facts about compact Hausdorff (CH) spaces: CH topologies on a given set are pairwise incomparible (one is not finer or coarser than the other). There ...
huurd's user avatar
  • 1,031
11 votes
0 answers
331 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
James Baxter's user avatar
  • 2,069
5 votes
2 answers
349 views

Codimension-1 subgroups of 3-manifold groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$. Do all ...
Gus's user avatar
  • 85
9 votes
1 answer
226 views

Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?

For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...
Taras Banakh's user avatar
  • 41.8k
1 vote
1 answer
313 views

Group action on quasi-isometric geodesic metric space [closed]

If a group $G$ acts on a geodesic metric space $X$, then does $G$ act on a geodesic metric space $Y$ which is quasi-isometric to $X$?
Anton's user avatar
  • 81
12 votes
0 answers
172 views

A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
Taras Banakh's user avatar
  • 41.8k
6 votes
1 answer
237 views

Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...
Alexander Gelbukh's user avatar