All Questions
Tagged with at.algebraic-topology gr.group-theory
40 questions
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
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
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
7
votes
1
answer
506
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
33
votes
3
answers
6k
views
(co)homology of symmetric groups
Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
- 1,589
21
votes
8
answers
4k
views
Cogroup objects
Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...
- 16k
4
votes
1
answer
394
views
$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?
I was trying to understand this interesting question by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
- 755
28
votes
4
answers
4k
views
Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,217
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.5k
20
votes
4
answers
3k
views
Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
- 373
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 ...
- 12.3k
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
2
votes
1
answer
359
views
Induced Map on Sp(2g,Z) is surjective
Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map ...
- 105
51
votes
1
answer
8k
views
What is Atiyah's topological formulation of the odd order theorem?
Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here).
During that year in Harvard, Thompson began his monumental ...
- 2,821
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 ...
- 63.9k
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.
- 29.1k
24
votes
1
answer
968
views
Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
- 11.9k
22
votes
6
answers
2k
views
Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
- 1,219
17
votes
1
answer
1k
views
A finite 2-group containing the dihedral group of order 16?
The dihedral group $D_{16}$ of order 16 has a presentation $D_{16}= \langle a,t \ | \ a^2=t^8=atat=e\rangle$.
Question: Does there exist a finite 2-group $G$ containing $D_{16}$ as a subgroup, and ...
- 11.1k
17
votes
1
answer
575
views
Group cochains invariant under the action of the symmetric group
Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
14
votes
2
answers
789
views
Restriction of a branched cover to its branch locus
Assume that we have a smooth, compact, complex surface $X$, and a smooth and irreducible divisor $B \subset X$. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G,...
- 66.3k
14
votes
2
answers
906
views
Acyclic group and finite CW-complex
Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
- 717
14
votes
1
answer
704
views
What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
The symmetric group $S_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real ...
- 54.6k
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
12
votes
2
answers
583
views
Do there exist acyclic simple groups of arbitrarily large cardinality?
Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...
- 63.9k
11
votes
9
answers
1k
views
Proving the impossibility of an embedding of categories
A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...
- 5,759
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
10
votes
1
answer
1k
views
Acyclic Finite Groups
A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also ...
- 2,630
8
votes
1
answer
672
views
Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?
I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me).
The cited question/answer ...
- 277
7
votes
2
answers
676
views
Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!) What can be said about subcomplexes of 2-complexes deformation retractible onto graphs.
I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
Is it true that the Euler characteristic of a finite connected aspherical ...
- 1,481
7
votes
2
answers
590
views
Pairs of Permutations up to Simultaneous Conjugation
The conjugacy classes of $S_n$ are the cycle types since if $\tau = (\dots)(\dots)\dots(\dots)$, the conjugation $\tau \mapsto \sigma \tau \sigma^{-1}$ permutes the labels in the cycles of $\tau$.
...
- 22.8k
7
votes
1
answer
2k
views
Automorphism group of the special unitary group $SU(N)$
Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...
- 2,147
7
votes
2
answers
860
views
mapping space between classifying spaces
I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces ...
- 1,974
6
votes
1
answer
406
views
Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
- 721
6
votes
2
answers
765
views
Cup products and the transfer map
Let $G_1$ be a finite-index subgroup of $G_2$. Let $i : H^{\ast}(G_2) \rightarrow H^{\ast}(G_1)$ be the induced map of rings. There is then a transfer homomorphism $\tau : H^{\ast}(G_1) \rightarrow ...
- 63
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
5
votes
2
answers
573
views
Are homotopy braid groups residually nilpotent?
A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...
- 1,108
5
votes
1
answer
384
views
Which groups have undetectable third U(1)-cohomology?
Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map
$$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
- 54.6k
4
votes
1
answer
625
views
Is there an algorithm for computing Schur multiplier?
Suppose we are given group $G=\langle a_1,\ldots,a_n \mid R_1=1,\ldots R_m=1 \rangle$. Is there an algorithm which computes a finite presentation for the Schur multiplier, i.e. second homology group $...
- 1,281
3
votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
0
votes
1
answer
143
views
Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755