Post on 05-Jan-2016
description
Simplicial structures on train tracks
Fedor Duzhin, Nanyang Technological University, Singapore
Plan of the talk
1. Braid groups
2. Crossed simplicial structure
3. Free groups and simplicial group structure on free groups
4. Combinatorial description of mapping classes
5. Simplicial structure on train tracks
Braid groups
A braid is: n descending strands joins {(i/(n-1),0,1)} to {(i/(n-1),0,0)} the strands do not intersect each other considered up to isotopy in R3
multiplication from top to bottom the unit braid
• =1 =
Artin’s presentationThe braid group on n strands Bn has the following presentation:
= = =
Braid groups on the sphere
The braid group Bn is then
Given any space M, its n-th ordered configuration space is
Obviously, the symmetric group Sn acts on F(M,n) by permuting coordinates.
The pure braid group Bn is
Braid group on the sphere Pure braid group on the sphere
Symmetric groups
Symmetric group Sn consists of bijections of an n-element set to itself
Presentation with generators - transpositions
Relations
Crossed simplicial structure
The braid group is a crossed simplicial group, that is,
Homomorphism to the permutation group
Face-operatorsDegeneracy-operators
Simplicial identities Crossed simplicial relation
Crossed simplicial structure
Face-operators are given by deleting a strand:
Crossed simplicial structure
Degeneracy-operators are given by doubling a strand:
Important result
Jon Berrick, Fred Cohen, Yan-Loi Wong, Jie Wu:There is a following exact sequence (actually, there are more)
Here the groups of Brunnian braids are
In other words, a braid is Brunnian if it becomes trivial after removing any its strand.
Free group Fn:Generators x0,x1,…,xn-1
No relations
Fn is the fundamental group of the n-punctured disk
AutFn is the group of automorphisms of Fn
Mapping class group consists of homotopy classes of self-homeomorphisms
x0 x1 xn-1
Free group
Also, Fn admits a simplicial group structure, that is,
Free group as a simplicial group
Face-operators(group homomorphisms)
Degeneracy-operators(group homomorphisms)
Simplicial identities:
Artin’s representation
Artin’s representation is obtained from considering braids as mapping classes
The disk is made of rubberPunctures are holesThe braid is made of wireThe disk is being pushed down along the braid
Theorem The braid group is isomorphic to the mapping class group of the punctured disk
Artin’s representation
Braids and general automorphisms are applied to free words on the right
Theorem (Artin)1. The Artin representation is
faithful2. The image of the Artin
representation is the set of automorphisms given by
where
satisfying
Permutative action
Theorem Generally, the braid groups act on free groups so that
commute for any braid a
In particular, for a pure braid a, the permutation πa is identity, so we have
Skeleton graphs
Let S be an n-punctured disk (or, generally, a surface with n punctures and k boundary components)
In order to give a combinatorial description to a mapping class (that is a self-homeomorphism of the surface S considered up to homotopy fixing the boundary of the disk pointwise and the set of punctures), one first defines a skeleton graph.
A skeleton graph is homotopy equivalent to the entire surface. It consists of n closed edges encircling punctures and a tree. Also, there are some natural equivalence relations. For example, one can remove a vertex of valence 1 or 2
Skeleton graphs
Given a homeomorphism f:S→S, the image of a skeleton graph is some other skeleton graph.
→f
Skeleton graphs
A map of a skeleton graph G to itself occurs as follows
inclusion
f
retraction
Skeleton graphs
Such a map induced on skeleton graphs is not a homeomorphismFor example, the following disk automorphism
induces graph map given by
bar meansreversed
Graph maps like this one are used in so calledtrain track algorithm (M. Bestvina, M. Handel)
Simplicial structure on train tracks
This is a current co-joint work with Jon Berrick and Jie Wu
Disclaimer: it’s not train tracks we construct simplicial structure on (train tracks will not even be defined in this talk)
We define a certain object called labelled skeleton graph. The set of labelled skeleton graphs is related to skeleton graph maps as
Skeleton graph maps Skeleton graph mapping classes
Labelled skeleton graphs Free group endomorphisms
Labelled skeleton graph
A labelled skeleton graphs looks like
Each edge is labelled by a free word
Closed edges are labelled by a permutation of the free generators
There are some equivalence relations
Simplicial structure on labelled skeleton graphs
Face-operator kills a closed edge (and applies the free group face operator to all labels)
face
Degeneracy-operator inserts two new edges
degeneracy
Thanks for your attention