All Questions
Tagged with at.algebraic-topology gr.group-theory
69 questions with no upvoted or accepted answers
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
15
votes
0
answers
716
views
Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
14
votes
0
answers
341
views
Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
- 141
14
votes
0
answers
414
views
Does the category of G-spectra know G?
I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
- 8,723
13
votes
0
answers
586
views
Finite groups inside an infinite group with the same homology
Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:
$G$ and $H$ are finite groups and $K$ is an infinite group.
there exist two monomorphisms $G \rightarrow K \leftarrow H$...
- 1,974
11
votes
0
answers
221
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 42.4k
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
11
votes
0
answers
203
views
Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
- 15.5k
10
votes
0
answers
458
views
is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
9
votes
0
answers
269
views
Colimits of symmetric groups
The infinite symmetric group $S_{\infty}$ of finitely supported permutations of $\mathbb{N}$ can be written as a colimit over the $S_n$'s with respect to the embedding $S_{n} \to S_{n+1}$ that maps $\...
- 7,637
9
votes
0
answers
372
views
Groups with trivial outer automorphism group and prescribed center?
Given an arbitrary abelian group $A$, can we find a group $G$ such that
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and
$Z(G)\simeq A$?
Why is this interesting? Given a group $G$, we have ...
- 27.2k
9
votes
0
answers
420
views
Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
- 163
9
votes
0
answers
439
views
(Torsion in) homology of free nilpotent groups
It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
- 4,600
9
votes
0
answers
376
views
Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
- 66.3k
8
votes
0
answers
128
views
What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?
Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
- 54.6k
8
votes
0
answers
238
views
Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?
Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...
- 54.6k
8
votes
0
answers
285
views
Are the braid groups good in the sense of Toën?
In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
- 1,635
8
votes
0
answers
204
views
Relationship between the p-radical subgroups and the parabolics in a BN-pair generality
A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
- 1,726
7
votes
0
answers
333
views
Positive instances of the Eilenberg-Ganea conjecture with families
The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
- 35.9k
7
votes
0
answers
422
views
Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups
I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...
- 1,108
6
votes
0
answers
484
views
Does a finitely generated aspherical group have an aspherical presentation with a finite generating set?
Let $G$ be a finitely generated group. Suppose $G$ has an aspherical presentation with a countably infinite generating set. Does $G$ have an aspherical presentation with a finite generating set?
Here ...
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
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
5
votes
0
answers
199
views
Outer and inner automorphism of $\mathrm{Pin}$ groups
$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
- 317
5
votes
0
answers
140
views
Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
- 5,405
5
votes
0
answers
636
views
Do the ternary braid groups arise in algebraic topology?
Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$
and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
5
votes
0
answers
222
views
Nilpotent Localization in Group Theory
Algebraic topologists have invented a very pretty technique of localizing nilpotent groups. (Garth Warner covers the topic in his book manuscript Topics in Topology and Homotopy Theory). For ...
- 6,870
4
votes
0
answers
453
views
Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
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 ...
4
votes
0
answers
320
views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.
Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,441
4
votes
0
answers
164
views
non-abelian tensor products of several groups
R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...
- 479
4
votes
0
answers
96
views
When are extensions of algebraically good groups algebraically good?
Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
- 1,635
4
votes
0
answers
136
views
Second homology of finitely presented group with free abelianisation
It is known that for a presented group $G=F/N$ we have
$$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$
In general, the right side seems to be difficult to calculate. I am in the special ...
- 1,623
4
votes
0
answers
172
views
from 2-cocycle to classifying map
Let $A,E,G:\mathrm{Set}_*\to\mathrm{Grp}_*$ be functors from pointed sets to (discrete) groups ($*=1$) together with natural transformations $i:A\to E, \ p: E\to G$ such that for any set $X$
\begin{...
4
votes
0
answers
239
views
The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group
Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
- 5,596
4
votes
0
answers
135
views
Exotic 2-adic lifts of mod $2$ Steinberg idempotent
Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...
- 3,051
4
votes
0
answers
144
views
When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?
Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
- 373
3
votes
0
answers
115
views
Finite homology of a homogeneous space
Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...
3
votes
0
answers
393
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
3
votes
0
answers
158
views
What is the meaning of local inertia conjugation property?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have:
Abstract. Let $\widehat{G T}^{1}$ ...
- 119
3
votes
0
answers
128
views
Salvetti complexes and cohomology of affine completion of Artin groups $E_6$ and $E_7$
After the solution of the Brieskorn-Arnold Pham conjecture on the asphericity of a space for affine Artin groups by Paolini and Salvetti MR4243019 (arXiv), I would like to know if there are ...
- 2,630
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
3
votes
0
answers
547
views
Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$
Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...
- 10.5k
3
votes
0
answers
208
views
Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$
Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.
(For example, given a short exact sequence
$$
1 \to BG_2 \to \mathbb{G} \to G_1 \to 1
$$
and the fiber sequence:
$$
B^2G_2 ...
- 10.5k
3
votes
0
answers
120
views
Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary
Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle
$\omega_3^G$ of a ...
- 10.5k
3
votes
0
answers
528
views
Classifying spaces
Note: this question was edited after a comment below
I'm reading into classifying spaces for the moment and I have some questions about these things. I'm using the following definition:
Given a ...
3
votes
0
answers
113
views
Have locally principal crossed homomorphisms been studied?
Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...
- 11.3k
3
votes
0
answers
257
views
Braids with an infinite number of strings
Has anyone developed a theory for braids with an infinite number of strings?
- 1,181
3
votes
0
answers
100
views
project limit on $n$- simplical complex which is principal homogeneous with respect to an action
The setting:
Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$
We suppose that we are given a system of ...
- 685
3
votes
0
answers
423
views
Cohomologies associated to residually torsion-free nilpotent groups
This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...
- 373