All Questions
Tagged with at.algebraic-topology gr.group-theory
288 questions
7
votes
1
answer
310
views
Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 3,948
10
votes
2
answers
337
views
Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
- 231
4
votes
1
answer
523
views
Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？ [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so，but I think it is possible on the N-dimensional torus.
- 12.9k
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 ...
- 12.9k
2
votes
1
answer
2k
views
Is a normal subgroup of a finitely presented group finitely generated or normal finitely generated?
Let $G$ be a finitely presented group and $N$ a normal subgroup. Is $N$ finitely generated or normally finitely generated? Here normally finitely generation means that for some finite set $S$ of ...
- 115
2
votes
0
answers
119
views
Crossed homomorphism as morphism in the ambient category
Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
- 199
29
votes
4
answers
3k
views
Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 63.9k
12
votes
2
answers
781
views
Where does the term "torsor" come from?
Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", ...
- 3,998
4
votes
0
answers
453
views
Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$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}.$$...
7
votes
1
answer
415
views
Classifying abelian (but non-central) group extensions using homotopy theory
Let $G$ be a group and let $A$ be an abelian group equipped with an action of $G$. Group extensions
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
inducing the ...
- 37.8k
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
2
votes
0
answers
84
views
Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
3
votes
1
answer
199
views
Subgroups of top cohomological dimension
Let $G$ be a geometrically finite group, i.e. there exists a finite CW complex of type $K(G,1)$.
By Serre's Theorem, every finite-index subgroup $H$ of $G$ satisfies $cd(H)=cd(G)$, but what about the ...
- 12.3k
6
votes
2
answers
437
views
Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
- 541
2
votes
0
answers
107
views
Homology functors and weak cofibers
I'm looking at a remark in the paper
Kainen, Paul C., "Weak Adjoint Functors", Mathematische Zeitschrift 122 (1971).
It is supposed to prove that generalized homology functors fail to ...
- 81
6
votes
1
answer
312
views
A Tate resolution for $\Sigma_p$ - Reference request
Below I will describe a mod $p$ Tate resolution for the symmetric group $\Sigma_p$, i.e. a $\mathbb{Z}$-graded periodic acyclic chain complex $C^*$ of finitely generated modules over $\mathbb{F}_p[\...
- 16.2k
4
votes
0
answers
425
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
1
vote
1
answer
279
views
Ways to prove that $n$-component Brunnian link is nontrivial
The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
- 28.2k
10
votes
2
answers
497
views
Equivariant cohomology of the complement to the arrangement $\bigcup_{i\neq j}\vec x_i = \vec x_j$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Conf{Conf}$Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidean) vector space over real numbers.
Let $G=\SO(V)$ be the ...
- 12.9k
16
votes
7
answers
2k
views
two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation.
Recently I've been reading J.P. May's A Concise Course in Algebraic Topology. In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate ...
- 21.3k
0
votes
0
answers
220
views
The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
7
votes
1
answer
519
views
When do covering spaces correspond to characteristic subgroups?
Given a covering space $p \colon X \to Y$, we get an injection $p^* \colon \pi_1(X) \to \pi_1(Y)$, and we know that the image $p^*(\pi_1(X))$ is normal in $\pi_1(Y)$ if an only if $p$ is regular, that ...
- 1,996
13
votes
3
answers
2k
views
Which groups are LERF?
A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
- 3,181
37
votes
1
answer
1k
views
If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?
$\DeclareMathOperator\Hom{Hom}$The question is in the title. If the isomorphism $\Hom(A, G) \cong \Hom(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely ...
- 12.9k
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 11.6k
3
votes
1
answer
248
views
Identifying group extension from cohomology class of $D_8$
I have the following problem. It is well known that $H^\ast(D_8,\mathbb{Z}/2)\cong \mathbb{F}_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). ...
- 11.2k
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 ...
- 63.9k
9
votes
1
answer
309
views
Comparing cohomology of a total complex with the cohomology of semidirect product
$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
- 28.7k
5
votes
1
answer
512
views
What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...
- 12.9k
27
votes
2
answers
796
views
Is there a flat manifold with trivial first homology?
Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
- 1,638
10
votes
4
answers
2k
views
Proving that a countable group is not finitely generated
I would like to learn about techniques for proving that a countable group is not finitely generated. I am also interested in learning about examples. Finally, I am particularly, but not exclusively, ...
- 19.6k
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
7
votes
2
answers
539
views
Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
- 9,580
11
votes
0
answers
221
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 42.4k
21
votes
8
answers
4k
views
Cogroup objects
Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...
- 4,713
1
vote
1
answer
249
views
Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
- 12.9k
14
votes
0
answers
341
views
Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
- 141
2
votes
0
answers
65
views
Representing elements of $H_2$ of a group using the bar (or standard) chain complex [duplicate]
Let $G$ be a discrete group, and let $a_1,b_1,\ldots,a_g,b_g \in G$ be elements such that
$$[a_1,b_1] \cdots [a_g,b_g] = 1$$
in $G$. These correspond to a map of a genus-$g$ surface group $\pi_1(\...
- 21
13
votes
2
answers
795
views
Which finite groups have low-degree essential cohomology?
Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
- 11.2k
1
vote
2
answers
334
views
Examples of finite polyhedra with finitely generated simple fundamental group
For $n\geq 2$, $P\mathbb{R}^n$ is a simple example of finite polyhedron with finitely generated simple fundamental group. I was wondering if someone could give me an example of a finite polyhedron ...
- 25k
4
votes
1
answer
207
views
Groups homology with coefficients fitting into filtration or exact sequence
Let $G$ be a group. I have two questions about the homology of $G$:
Consider a finite exact sequence
$$0 \rightarrow M_1 \rightarrow \cdots \rightarrow M_m \rightarrow 0$$
of $G$-modules. How are ...
CommunityBot
- 1
0
votes
0
answers
194
views
Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
- 63.9k
0
votes
0
answers
378
views
Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
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 (...
12
votes
2
answers
1k
views
Cohomological dimension of a homomorphism
Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that $...
- 35.9k
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
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 $\widehat{G T}^{1}$ ...
- 63.9k
6
votes
1
answer
316
views
Groups with unusual cohomological dimension of direct product
$\DeclareMathOperator\cd{cd}$Are there any known examples of non-free groups with a property that $\cd(G)+1 = \cd(G \times G)$, or, less restrictive, $G, H$ with $\cd \neq 1, \infty$ such that $\cd(H)+...
- 3,451