All Questions
Tagged with gn.general-topology gr.group-theory
124 questions
3
votes
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 $\...
- 48.8k
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 $G$...
- 361
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 ...
- 4,081
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 ...
- 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 $B/A$. ...
- 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 ...
- 1
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
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 ...
- 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 \...
- 661
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 ...
- 883
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) ...
- 253
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 ...
- 356
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 ...
- 253
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 ...
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 ...
- 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
$\...
- 2,221
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 ...
- 8,502
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 ...
- 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 ...
- 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 ...
- 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 -- ...
- 19.8k
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 $...
- 603
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 ...
- 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 ...
- 3,011
4
votes
2
answers
381
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 ...
- 7,527
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 ...
- 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 ...
- 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 "...
- 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 ...
- 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 ...
- 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}...
- 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?
- 4,786
9
votes
2
answers
901
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 ...
- 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 ...
- 3,317
3
votes
2
answers
299
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 ...
- 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\...
- 63.9k
16
votes
1
answer
501
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 $G \curvearrowright X$ ...
- 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 ...
- 12.2k
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 ...
- 1,150
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 ...
- 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?
- 451
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, ...
- 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 ...
- 998
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 ...
- 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?
...
- 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 ...
- 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\...
- 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 $X$?
- 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 $\...
- 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 ...
