353 research outputs found
A family of representations of braid groups on surfaces
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.Comment: 21 pages, 4 figure
A note on the Lawrence-Krammer-Bigelow representation
A very popular problem on braid groups has recently been solved by Bigelow
and Krammer, namely, they have found a faithful linear representation for the
braid group B_n. In their papers, Bigelow and Krammer suggested that their
representation is the monodromy representation of a certain fibration. Our goal
in this paper is to understand this monodromy representation using standard
tools from the theory of hyperplane arrangements. In particular, we prove that
the representation of Bigelow and Krammer is a sub-representation of the
monodromy representation which we consider, but that it cannot be the whole
representation.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-24.abs.htm
Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials
In this paper, we study fundamental groups of strata of the moduli space of
quadratic differentials. We use certain properties of the Abel-Jacobi map,
combined with local surgeries on quadratic differentials, to construct quotient
groups of the fundamental groups for a particular family of strata.Comment: 43 pages, 7 figures. Version 2: Minor typos fixed, Section 3 removed
and may now be found in arXiv:0804.043
Modification rule of monodromies in R_2-move
An R_2-move is a homotopy of wrinkled fibrations which deforms images of
indefinite fold singularities like Reidemeister move of type II. Variants of
this move are contained in several important deformations of wrinkled
fibrations, flip and slip for example. In this paper, we first investigate how
monodromies are changed by this move. For a given fibration and its vanishing
cycles, we then give an algorithm to obtain vanishing cycles in one reference
fiber of a fibration, which is obtained by applying flip and slip to the
original fibration, in terms of mapping class groups. As an application of this
algorithm, we give several examples of diagrams which were introduced by
Williams to describe smooth 4-manifolds by simple closed curves of closed
surfaces.Comment: 34 pages, 15 figure
Loop homology of spheres and complex projective spaces
In his Inventiones paper, Ziller (Invent. Math: 1-22, 1977) computed the
integral homology as a graded abelian group of the free loop space of compact,
globally symmetric spaces of rank 1. Chas and Sullivan (String Topology,
1999)showed that the homology of the free loop space of a compact closed
orientable manifold can be equipped with a loop product and a BV-operator
making it a Batalin-Vilkovisky algebra. Cohen, Jones and Yan (The loop homology
algebra of spheres and projective spaces, 2004) developed a spectral sequence
which converges to the loop homology as a spectral sequence of algebras. They
computed the algebra structure of the loop homology of spheres and complex
projective spaces by using Ziller's results and the method of Brown-Shih (Ann.
of Math. 69:223-246, 1959, Publ. Math. Inst. Hautes \'Etudes Sci. 3: 93-176,
1962). In this note we compute the loop homology algebra by using only spectral
sequences and the technique of universal examples. We therefore not only obtain
Zillers' and Brown-Shihs' results in an elementary way, we also replace the
roundabout computations of Cohen, Jones and Yan (The loop homology algebra of
spheres and projective spaces, 2004) making them independent of Ziller's and
Brown-Shihs' work. Moreover we offer an elementary technique which we expect
can easily be generalized and applied to a wider family of spaces, not only the
globally symmetric ones.Comment: 10 pages, 8 figure
Estimating the higher symmetric topological complexity of spheres
We study questions of the following type: Can one assign continuously and
-equivariantly to any -tuple of distinct points on the sphere
a multipath in spanning these points? A \emph{multipath} is a
continuous map of the wedge of segments to the sphere. This question is
connected with the \emph{higher symmetric topological complexity} of spheres,
introduced and studied by I. Basabe, J. Gonz\'alez, Yu. B. Rudyak, and D.
Tamaki. In all cases we can handle, the answer is negative. Our arguments are
in the spirit of the definition of the Hopf invariant of a map by means of the mapping cone and the cup product.Comment: This version has minor corrections compared to what published in AG
Planar open books with four binding components
We study an explicit construction of planar open books with four binding
components on any three-manifold which is given by integral surgery on three
component pure braid closures. This construction is general, indeed any planar
open book with four binding components is given this way. Using this
construction and results on exceptional surgeries on hyperbolic links, we show
that any contact structure of S^3 supports a planar open book with four binding
components, determining the minimal number of binding components needed for
planar open books supporting these contact structures. In addition, we study a
class of monodromies of a planar open book with four binding components in
detail. We characterize all the symplectically fillable contact structures in
this class and we determine when the Ozsvath-Szabo contact invariant vanishes.
As an application, we give an example of a right-veering diffeomorphism on the
four-holed sphere which is not destabilizable and yet supports an overtwisted
contact structure. This provides a counterexample to a conjecture of Honda,
Kazez, Matic from arXiv:0609734 .Comment: 19 pages, 10 figures. Two cancelling sign errors remove
Optimal bounds for a colorful Tverberg--Vrecica type problem
We prove the following optimal colorful Tverberg-Vrecica type transversal
theorem: For prime r and for any k+1 colored collections of points C^l of size
|C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color
classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the
collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane
that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k)
is even or k=0.
Along the proof we obtain three results of independent interest: We present
two alternative proofs for the special case k=0 (our optimal colored Tverberg
theorem (2009)), calculate the cohomological index for joins of chessboard
complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant
bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).Comment: Substantially revised version: new notation, improved results,
additional references; 12 pages, 2 figure
Knaster's problem for almost -orbits
In this paper some new cases of Knaster's problem on continuous maps from
spheres are established. In particular, we consider an almost orbit of a
-torus on the sphere, a continuous map from the sphere to the real
line or real plane, and show that can be rotated so that becomes
constant on
The center of some braid groups and the Farrell cohomology of certain pure mapping class groups
In this paper we first show that many braid groups of low genus surfaces have
their centers as direct factors. We then give a description of centralizers and
normalizers of prime order elements in pure mapping class groups of surfaces
with spherical quotients using automorphism groups of fundamental groups of the
quotient surfaces. As an application, we use these to show that the -primary
part of the Farrell cohomology groups of certain mapping class groups are
elementary abelian groups. At the end we compute the -primary part of the
Farrell cohomology of a few pure mapping class groups.Comment: 16 page
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