709 research outputs found
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Khovanov defined graded homology groups for links L in R^3 and showed that
their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's
construction does not extend in a straightforward way to links in I-bundles M
over surfaces F not D^2 (except for the homology with Z/2 coefficients only).
Hence, the goal of this paper is to provide a nontrivial generalization of his
method leading to homology invariants of links in M with arbitrary rings of
coefficients. After proving the invariance of our homology groups under
Reidemeister moves, we show that the polynomial Euler characteristics of our
homology groups of L determine the coefficients of L in the standard basis of
the skein module of M. Therefore, our homology groups provide a
`categorification' of the Kauffman bracket skein module of M. Additionally, we
prove a generalization of Viro's exact sequence for our homology groups.
Finally, we show a duality theorem relating cohomology groups of any link L to
the homology groups of the mirror image of L.Comment: Version 2 was obtained by merging math.QA/0403527 (now removed) with
Version 1. This version is published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-52.abs.htm
Conformally Einstein Products and Nearly K\"ahler Manifolds
In the first part of this note we study compact Riemannian manifolds (M,g)
whose Riemannian product with R is conformally Einstein. We then consider
compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the
Gray--Hervella classification admitting a parallel vector field and show that
(under some regularity assumption) they are obtained as mapping tori of
isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular,
we obtain examples of inhomogeneous locally (non-globally) conformal nearly
K\"ahler compact manifolds
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