Configuration Spaces and Braid Groups (original) (raw)

On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

The Quarterly Journal of Mathematics

The mapping class group Gammak(Ng)\Gamma ^k(N_g)Gammak(Ng) of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of mathbbRtextrmP2{\mathbb{R}} \textrm{P}^2mathbbRtextrmP2, we analyze the Serre spectral sequence of a fiber bundle $ F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3) $ where XkX_kXk is a K(Gammak(mathbbRtextrmP2),1)K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)K(Gammak(mathbbRtextrmP2),1) and Fk(mathbbRtextrmP2)/SigmakF_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _kFk(mathbbRtextrmP2)/Sigmak denotes the configuration space of unordered kkk-tuples of distinct points in mathbbRtextrmP2{\mathbb{R}} \textrm{P}^2mathbbRtextrmP2. As a consequence, we express the mod-2 cohomology of Gammak(mathbbRtextrmP2)\Gamma ^k({\mathbb{R}} \textrm{P}^2)Gammak(mathbbRtextrmP2) in terms of that of Fk(mathbbRtextrmP2)/SigmakF_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _kFk(mathbbRtextrmP2)/Sigmak.

Braid groups and discrete diffeomorphisms of the punctured disk

Mathematische Zeitschrift, 2017

We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding Br 2g+2 ↪ Mod g,2 to a group homomorphism between diffeomorphism groups.

TOPOLOGICAL STEPS TOWARD THE HOMFLYPT SKEIN MODULE OF THE LENS SPACES L(p, 1) VIA BRAIDS

In this paper we work toward the Homflypt skein module of the lens spaces L(p, 1), S(L(p, 1)), using braids. In particular, we establish the connection between S(ST), the Homflypt skein module of the solid torus ST, and S(L(p, 1)) and arrive at an infinite system, whose solution corresponds to the computation of S(L(p, 1)). We start from the Lambropoulou invariant X for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, Λ, of S(ST) presented in [DL1]. We show that S(L(p, 1)) is obtained from S(ST) by considering relations coming from the performance of braid band moves (bbm) on elements in the basis Λ, where the braid band moves are performed on any moving strand of each element in Λ. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set Λ. The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. 3-manifolds, since any 3-manifold can be obtained by surgery on S^3 along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.

The mod 2 equivariant cohomology algebras of configuration spaces

Pacific Journal of Mathematics, 1990

The mod 2 equivariant cohomology algebras of configuration spaces are determined by means of the Dickson characteristic classes, which are derived from the modular invariants of the general linear groups GL(« , Z 2) and closely related to the Stiefel-Whitney classes.

Braid groups in complex projective spaces

Advances in Geometry, 2012

We describe the fundamental groups of ordered and unordered k−point sets in CP n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of points.

Homotopy theory of non-orientable mapping class groups

2012

We give a homotopical approach to the theory of mapping class groups of surfaces with marked points. Using conguration spaces we construct Eilenberg-MacLane spaces K(; 1) for the mapping class groups of the projective plane P 2 and the Klein bottle IK. These spaces are closely related to theK(; 1) spaces for the corresponding braid groups. Some cohomological consequences of this approach are presented.

Abstract commensurators of braid groups

Journal of Algebra, 2006

Let B n be the braid group on n ≥ 4 strands. We show that the abstract commensurator of B n is isomorphic to Mod(S) ⋉ (Q × ⋉ Q ∞ ), where Mod(S) is the extended mapping class group of the sphere with n + 1 punctures.

A Family of representations of braid groups on surfaces

2010

We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.