All Questions
Tagged with gn.general-topology gr.group-theory
116
questions
2
votes
2
answers
141
views
$String/CP^{\infty}=Spin$ or a correction to this quotient group relation
We know that there is a fiber sequence:
$$
... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ...
$$
Is this fiber sequence induced from a short exact sequence?
If so, is that
$$
1 \to B^2 Z = B S^...
- 327
5
votes
0
answers
234
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 619
0
votes
0
answers
58
views
A cellular automaton with an image that is not closed
Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
- 885
8
votes
1
answer
138
views
Stone-topological/profinite equivalence for quandles
A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...
- 253
0
votes
0
answers
94
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 280
4
votes
1
answer
176
views
Profinite groups with isomorphic proper, dense subgroups are isomorphic
I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
- 253
4
votes
0
answers
420
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
5
votes
1
answer
214
views
Extreme amenability of topological groups and invariant means
Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
- 125
0
votes
0
answers
100
views
Classification of closures of additive subgroups of $\mathbb{R}^n$
If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either
$\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or
$\...
- 2,209
3
votes
1
answer
147
views
Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
- 8,322
4
votes
0
answers
74
views
Is each TS-topologizable group TG-topologizable?
Definition 1. A topology $\tau$ on a group $X$ is called
$\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$;
$\bullet$ a ...
- 40.2k
7
votes
0
answers
137
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$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 ...
- 40.2k
0
votes
0
answers
134
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 ...
- 1,039
6
votes
1
answer
462
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 -- ...
- 19.4k
2
votes
0
answers
213
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 $...
- 593
1
vote
1
answer
233
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
387
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 ...
- 2,672
4
votes
2
answers
319
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 ...
- 6,567
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
266
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 ...
- 1,916
2
votes
1
answer
190
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 "...
- 13.9k
-7
votes
1
answer
335
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 ...
- 25
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
130
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
437
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?
- 4,674
9
votes
2
answers
765
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 ...
- 2,360
9
votes
2
answers
496
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 ...
- 2,993
3
votes
2
answers
238
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
13
votes
1
answer
412
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\...
- 59.4k
15
votes
1
answer
474
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 $G \curvearrowright X$ ...
- 4,010
15
votes
1
answer
479
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 ...
- 9,138
4
votes
1
answer
252
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 ...
- 1,131
11
votes
1
answer
911
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 ...
- 2,993
-2
votes
1
answer
128
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
80
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, ...
- 25.3k
5
votes
0
answers
272
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 ...
- 923
5
votes
2
answers
303
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 ...
- 935
11
votes
0
answers
322
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?
...
- 2,029
5
votes
2
answers
310
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
225
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\...
- 40.2k
1
vote
1
answer
291
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
171
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 $\...
- 40.2k
6
votes
1
answer
220
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 ...
1
vote
0
answers
116
views
A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
- 1,807
2
votes
1
answer
74
views
Haar-$\mathcal{I}$ set and Polish groups
Let $\mathcal{I}$ be a semi-ideal of sets with empty interior on a compact metrizable space $K$. Let an $F_σ$-set $A$ in a Polish group $X$ generically Haar-$\mathcal{I}$.
Then is $A$ always ...
- 1,569
2
votes
1
answer
140
views
Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group
Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous ...
- 6,050
14
votes
1
answer
285
views
Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group?
Let $S_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt_\omega$ be the subgroup of $S_\omega$ consisting of even permutations of $\omega$ (i.e., the ...
- 40.2k
5
votes
2
answers
430
views
Is each locally compact group topology on the permutation group discrete?
Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete?
Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
- 40.2k
2
votes
2
answers
151
views
How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]
How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...
1
vote
0
answers
108
views
Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
- 1,807