All Questions
Tagged with at.algebraic-topology gr.group-theory
288 questions
85
votes
23
answers
11k
views
Solving algebraic problems with topology
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...
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
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
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 ...
- 1,141
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
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
32
votes
3
answers
2k
views
Is the Hurewicz theorem ever used to compute abelianizations?
The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$. This gives a potential method for computing the ...
user81684
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
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
29
votes
2
answers
1k
views
Quillen + construction for finite groups
Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?
- 1,629
28
votes
2
answers
6k
views
What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\...
- 5,534
28
votes
4
answers
4k
views
Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,217
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
25
votes
3
answers
2k
views
Are fundamental groups of aspherical manifolds Hopfian?
A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth ...
- 32.4k
24
votes
3
answers
4k
views
Subgroups of free abelian groups are free: a topological proof?
There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which ...
- 65.4k
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
23
votes
9
answers
4k
views
What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
- 1,861
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
22
votes
2
answers
1k
views
The image of the point-pushing group in the hyperelliptic representation of the braid group
Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which ...
- 19.2k
22
votes
1
answer
719
views
What is the cohomological dimension of the commutator subgroup of the pure braid group?
I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature.
Because $[P_n,P_n]$ ...
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
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
21
votes
5
answers
1k
views
Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
- 15.1k
21
votes
2
answers
622
views
Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
- 2,479
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
20
votes
3
answers
1k
views
Center of a simply-connected simple compact Lie group and McKay correspondence
Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...
- 3,051
18
votes
2
answers
592
views
primary decomposition for nonabelian cohomology of finite groups
Let $G$ be a finite group, and let $M$ be a group on which $G$ acts (via a homomorphism $G\to \operatorname{Aut}(M)$).
If $M$ is abelian, hence a $\mathbb{Z}G$-module, there is a primary ...
- 35.9k
18
votes
2
answers
1k
views
Fundamental group of punctured simply connected subset of $\mathbb{R}^2$
(This question is originally from Math.SE where it was suggested that I ask the question here)
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...
- 2,889
18
votes
1
answer
2k
views
Grothendieck's question on the Brauer group for groups
Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear ...
- 5,062
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
17
votes
1
answer
998
views
Where should I search for computations of group cohomology rings of not-too-complicated finite groups?
A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
...
- 6,881
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
683
views
Relationship between Smith's special homology groups and equivariant homology theory
EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is ...
- 44.8k
17
votes
3
answers
1k
views
The second homotopy group of a simple CW-complex
Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?
- 1,181
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
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 ...
- 6,069
16
votes
1
answer
505
views
How many cells needed to build the classifying space $BG$?
Let $G$ be a finitely presented group of cohomological dimension $n$.
Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional ...
- 11.9k
15
votes
5
answers
3k
views
When are all centralizers in a Lie group connected?
Let $G$ be a compact connected Lie group acting on itself by conjugation,
$$ G\times G\to G,\qquad (\sigma,h)\mapsto \sigma h \sigma^{-1}.$$
The fixed point set of a closed subgroup $H\le G$ equals ...
- 35.9k
15
votes
3
answers
14k
views
How to demonstrate $SO(3)$ is not simply connected?
A quote from Wikipedia's article on the Rotation group:
Consider the solid ball in $\mathbb{R}^3$ of
radius $\pi$ [...].
Given the above, for every point in
this ball there is a rotation, ...
- 153
15
votes
2
answers
968
views
Semidirect product decomposition of the Borromean rings group
Let $X=S^3\setminus B$ be the link complement of the Borromean rings.
(source)
Then $G=\pi_1(X)$ has a presentation of the form
$$
G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; ...
- 35.9k
15
votes
3
answers
926
views
Lower central series quotients in terms of (co)homology
Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...
- 7,377
15
votes
1
answer
640
views
Torsion-free group that is not of type F but is virtually of type F
Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...
- 153
15
votes
1
answer
629
views
Characteristic classes of symmetric group $S_4$
For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
- 439
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
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
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
3
answers
683
views
Compact manifolds with big mapping class group
I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group.
Are there higher dimensional manifolds (which are not in some way reducible to ...
- 2,696
14
votes
2
answers
416
views
Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$
This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...
- 183
14
votes
2
answers
1k
views
Are acyclic subcomplexes of finite contractible 2-complexes contractible?
Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi_1(X)$ trivial?)...
- 1,481
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
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