All Questions
Tagged with at.algebraic-topology gr.group-theory
288 questions
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 ...
CommunityBot
- 1
3
votes
2
answers
699
views
What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?
I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...
CommunityBot
- 1
2
votes
1
answer
387
views
Generators of the colored braid group (two colors), reference
I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white.
It is easy to find a set of generators for $B_{n,n}$:
$$
\begin{cases}
\...
- 4,959
3
votes
0
answers
100
views
project limit on $n$- simplical complex which is principal homogeneous with respect to an action
The setting:
Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$
We suppose that we are given a system of ...
- 685
8
votes
1
answer
1k
views
Almost-direct product and 1-formality
Let $G$ be a finitely presented group. To $G$ is associated in a functorial way a Malcev Lie algebra which can be constructed in several equivalent ways. Roughly speaking, it is the quotient of the ...
- 1,960
20
votes
3
answers
1k
views
Center of a simply-connected simple compact Lie group and McKay correspondence
Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...
- 3,051
5
votes
1
answer
428
views
Centralizers in the universal central extensions of the alternating groups?
For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
CommunityBot
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7
votes
0
answers
422
views
Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups
I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...
- 1,108
3
votes
1
answer
445
views
Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)
What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...
- 4,959
5
votes
2
answers
666
views
HNN extensions which are free products
which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
- 345
1
vote
0
answers
275
views
Explicitly showing that a free group is LERF [closed]
Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...
- 11.3k
3
votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
4
votes
2
answers
337
views
A Karrass-Solitar theorem for surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...
- 496
5
votes
1
answer
264
views
Bases of surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
CommunityBot
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8
votes
1
answer
382
views
Second homology of mapping class group of genus 3
In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the ...
- 12.8k
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 ...
- 8,717
5
votes
1
answer
964
views
1D TQFT in Freed-Hopkins-Lurie-Teleman
In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory.
$F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual.
$F(\circ-\circ)$ is ...
- 118k
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}*\...
- 66.3k
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
1
vote
1
answer
151
views
A formula for isotropy group $\pi_1(G_a)$
Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
- 4,920
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
5
votes
1
answer
629
views
What is known about maximal free subgroups of surface groups?
Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...
- 68.9k
11
votes
1
answer
620
views
Is $SL(n,\mathbb{Z})$ a CAT(0) group?
Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.
- 45k
9
votes
1
answer
202
views
Are compact simple groups homotopically non-abelian?
Take a compact connected simple centreless Lie group $G$.
Can the commutator map $G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?
I am interested mostly in the case, ...
CommunityBot
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8
votes
2
answers
632
views
Are torus knot groups linear?
The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful ...
CommunityBot
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1
vote
3
answers
553
views
Homomorphisms of the free group $F_n$ to $GL_k(\mathbb{R})$ [closed]
Is there something known about the set of all homomorphisms from the free group on $n$ generators $F_n$ to the real general linear group $GL_k(\mathbb{R})$ modulo conjugation, i.e.
$$
Hom(F_n, GL_k(\...
- 6,210
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 ...
- 68.9k
3
votes
1
answer
429
views
Is a retract of a group of type F_n again of this type?
It has been asked here, whether a retract of a finitely presented group is again finitely presented, i.e. if $G$ is a finitely presented group and $H$ is a group which fits into a split exact sequence ...
CommunityBot
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7
votes
1
answer
433
views
Associativity with infinite nesting
I was trying to understand the Eilenberg-Mazur swindle (which I learned about here) especially as it could be used to show that if $A, B$ are compact (topological) $n$-manifolds whose connect sum is $...
CommunityBot
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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?)...
CommunityBot
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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
2
votes
1
answer
179
views
A sort of "group-ring" construction on coefficient systems in group homology (+ special case involving GL(n,Z))
Let $G$ be a discrete group and $M$ be an $RG$-module for some ring $R$ (I'm happy to assume that $R = \mathbb{Q}$). Define $R[M]$ to be the set of $R$-linear combinations of formal symbols of the ...
- 35.2k
10
votes
2
answers
890
views
Are virtual cubulated groups cubulated?
Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex?
Edit: After ...
- 15.4k
1
vote
1
answer
177
views
Intersections of subgroups of surface groups [closed]
Let $\mathcal{S}_g$ denote the fundamental group of an oriented surface of genus $g\ge 2$.
Does $\mathcal{S}_g$ contain subgroups $A$ and $B$ of finite index such that $A\cap B = \lbrace e\rbrace$?
- 25k
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 ...
- 30.4k
10
votes
1
answer
580
views
Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups
I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions.
$G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ (...
- 10.6k
1
vote
0
answers
112
views
When are graphs of cohomologically complete groups cohomologically complete?
A group $G$ is cohomologically $p$- complete if the canonical map from $G$ to it's pro$-p$ completion $\hat G^p$ induces an isomorphism on cohomology $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow ...
- 11
3
votes
1
answer
426
views
Naturality of the transfer in group cohomology
Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module ...
- 18.2k
14
votes
4
answers
677
views
Computing homotopy groups of X such that pi_1(X) has solvable word problem
The paper
E. H. Brown, Jr., Finite computability of Postnikov complexes, Ann. of Math. (2) 65 (1957), 1-20.
proves that if $X$ is a finite simply-connected simplicial complex, then there is an ...
- 12.3k
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 ...
- 5,546
8
votes
1
answer
532
views
Localizations of non-nilpotent spaces
For simplicity let's talk about $p$-localizations of spaces for a fixed prime $p$. Every space $X$ has a well-defined $p$-localization which can be constructed by the small object argument and which ...
- 9,870
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 $\tau$.
...
- 50.8k
6
votes
2
answers
447
views
Why additional constraint is need for this two groups to be isomorphic?
I'm reading AMS's book Papers on Topology, which collects Poincare's papers on topology.
However, the first paper stops me.
In the paper, he considered the group generated by transformations in $\...
- 96.4k
6
votes
3
answers
1k
views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
CommunityBot
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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
2
votes
1
answer
261
views
How many quotients can a finitely generated group have or how many bundles over aspherical spaces does a fixed total space support?
Consider $M^3_{pq}$,
a torus bundle over $S^1$ with fundamental group the HNN extension generated by three generators $x,y,z$ satisfying the relations $\quad [x,y], \quad x^z = x^p \quad$ and $y^z = y^...
CommunityBot
- 1
7
votes
2
answers
672
views
cohomological dimension of a group acting on a product
I recently came across an interesting result of Kobayashi [Corollary 5.5], a special case of which is the following: Suppose $\Gamma$ is a discrete torsion free subgroup of $SL_n(\mathbb{R})$ which ...
- 16.2k
10
votes
1
answer
855
views
finite complex with non-finitely generated homology with local coefficients
I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a ...
- 25k
6
votes
1
answer
309
views
For which rings R is SL_n(R) a virtual duality group
A famous theorem of Borel and Serre says that if $R$ is the ring of integers in an algebraic number field, then $\text{SL}_n(R)$ satisfies virtual Bieri-Eckmann duality. In other words, there exists ...
- 16.2k
10
votes
4
answers
6k
views
Commutativity of the fundamental group of any Lie Group [closed]
How do we formally prove that the fundamental group of any Lie group is always commutative?
CommunityBot
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