All Questions
Tagged with at.algebraic-topology gr.group-theory
288 questions
3
votes
1
answer
233
views
Aspherical amalgamations without injective maps
The situation I find myself in is as follows: I have a CW complex $X$ which is covered by two subcomplexes $A$ and $B$ and I know that $A$, $B$ and $A \cap B$ are connected and aspherical. The term ...
- 1,680
5
votes
3
answers
411
views
Computability and complexity of computing $|Hom(G,H)|$ for finitely presented groups G, H.
In the general case, I want to say that determining $|Hom(G,H)|$ is incomputable, arguing that you could use the number to test for simplicity of a presentation, but I am new to this area and I keep ...
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
7
votes
2
answers
957
views
Computations in group cohomology
Hello,
Given a finitely presentable group $G$, I'm interested in the cup-product from $H^1$ to $H^2$ with real coefficients. I want to know if this is explicitly computable (with a computer) with a ...
- 339
1
vote
1
answer
156
views
Sufficient Conditions for Free Indecomposability
An interesting fact was relayed to me in another question of mine that
If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi_1(M)$ is freely ...
- 726
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
5
votes
2
answers
754
views
explicit linear representations of fundamental groups of surfaces
I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix ...
- 1,050
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(\...
- 471
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}*\...
- 5,509
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
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 $...
- 2,299
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
7
votes
1
answer
859
views
How commutative is Quillen's Plus-Construction?
This question is inspired by this question about the dependence of K-theory on the order of multiplication in the ring. I did not think long about it, so maybe the answer really lies on the surface; ...
- 25.5k
3
votes
2
answers
1k
views
presentations of the trivial group
I just came across this statement in Bowditch's notes on geometric group theory that $\langle a,b\ |\ aba^{-1}b^{-2},a^{-2}b^{-1}ab \rangle$ is a presentation of the trivial group. Does anyone know if ...
- 31
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
-1
votes
1
answer
421
views
How does the discrete group act on simplicial set level by level
Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?
For example, if ...
- 681
3
votes
3
answers
1k
views
Another group cohomology cup product question
I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following:
Let $G=F/R$ be a finitely presented group ...
- 1,422
8
votes
2
answers
596
views
Infinite loop space maps into or out of BAut(F_n)
There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become ...
- 2,734
7
votes
2
answers
537
views
Residually finite + torsion free + finite index = finite complex?
Suppose $G$ is a residually-finite group and $H < G$ a torsion-free subgroup of finite index.
What characterizes such $G$ such that $BH$ is homotopic to a finite complex?
I believe Serre showed ...
- 2,734
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
8
votes
1
answer
1k
views
Beyond an intro to topological graph theory...
I'm looking to find out what active areas of research there are in topological graph theory, particularly those that interface strongly with other areas of math (say, group theory, algebraic topology, ...
- 1,180
2
votes
2
answers
617
views
Comparing lower central series and augmentation ideal completions
Let G be a group. Let $G^p$ be the completion of G with respect to the mod p lower central series of G.i.e. $G^p=\varprojlim_{q} G/\gamma_qG$, where $\gamma_qG$ is generated by all $\{[x_1,\cdots,x_s]^...
user2529
11
votes
2
answers
843
views
covers of $Z^\infty$
Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
user6976
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
10
votes
1
answer
635
views
Self-homomorphisms of surface groups
Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?
- 932
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
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 ...
- 1,373
3
votes
3
answers
6k
views
Homology of Surfaces with Holes
The classification theorem for surfaces says that the complete set of homeomorphism classes of surfaces is
{ $S_g : g \geq 0$ } $ \cup$ { $N_k : k \geq 1$ },
where $S_g$ is a sphere with $g$ ...
- 32.1k
7
votes
1
answer
672
views
How does this geometric description of the structure of PSL(2, Z) actually work?
There is a beautiful way to see that the congruence subgroup $\Gamma(2)$ is free on two generators: the action of $\Gamma(2)$ on $\mathbb{H}$ is free and properly discontinuous, and there is a modular ...
- 118k
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
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
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
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
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
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
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
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