All Questions
14 questions
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
5
votes
0
answers
636
views
Do the ternary braid groups arise in algebraic topology?
Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$
and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
6
votes
2
answers
331
views
Epimorphisms from the genus $2$ surface braid group to finite groups
This question is somehow related to my previous MO question Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$; for the reader convenience, let me write down again the relevant ...
- 66.3k
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 $[P_n,P_n]$ ...
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
3
votes
0
answers
257
views
Braids with an infinite number of strings
Has anyone developed a theory for braids with an infinite number of strings?
- 1,181
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}
\...
- 5,063
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
9
votes
2
answers
650
views
Action of the homotopy braid groups on reduced free groups
Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...
- 1,108
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
11
votes
1
answer
811
views
What is an interpretation of the relation in the cohomology of the pure braid groups?
In 1968, Arnol'd proved that the integral cohomology of the pure braid group $P_n$ is isomorphic to the exterior algebra generated by the collection of degree-one classes $\omega_{i,j}\ (1 \le i < ...
- 2,830
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
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 ...
- 8,524
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