53 research outputs found
On homotopy invariants of combings of 3-manifolds
Combings of oriented compact 3-manifolds are homotopy classes of nowhere zero
vector fields in these manifolds. A first known invariant of a combing is its
Euler class, that is the Euler class of the normal bundle to a combing
representative in the tangent bundle of the 3-manifold . It only depends on
the Spin-structure represented by the combing. When this Euler class is a
torsion element of , we say that the combing is a torsion combing.
Gompf introduced a -valued invariant of torsion combings of
closed 3-manifolds that distinguishes all combings that represent a given
Spin-structure. This invariant provides a grading of the Heegaard Floer
homology for manifolds equipped with torsion Spin-structures. We
give an alternative definition of the Gompf invariant and we express its
variation as a linking number. We also define a similar invariant for
combings of manifolds bounded by . We show that the -invariant,
that is the simplest configuration space integral invariant of rational
homology spheres, is naturally an invariant of combings of rational homology
balls, that reads where is the
Casson-Walker invariant. The article also includes a mostly self-contained
presentation of combings.Comment: 31 pages + 1 page at the end that summarizes the changes with respect
to the first versio
An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations
These introductory lectures show how to define finite type invariants of
links and 3-manifolds by counting graph configurations in 3-manifolds,
following ideas of Witten and Kontsevich. The linking number is the simplest
finite type invariant for 2-component links. It is defined in many equivalent
ways in the first section. As an important example, we present it as the
algebraic intersection of a torus and a 4-chain called a propagator in a
configuration space. In the second section, we introduce the simplest finite
type 3-manifold invariant, which is the Casson invariant (or the
Theta-invariant) of integer homology 3-spheres. It is defined as the algebraic
intersection of three propagators in the same two-point configuration space. In
the third section, we explain the general notion of finite type invariants and
introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we
sketch an original construction based on configuration space integrals of
universal finite type invariants for links in rational homology 3-spheres and
we state open problems. Our construction generalizes the known constructions
for links in the ambient space, and it makes them more flexible. In Section 6,
we present the needed properties of parallelizations of 3-manifolds and
associated Pontrjagin classes, in details.Comment: 68 pages. Change of title, updates and minor reorganization of notes
of five lectures presented in the ICPAM-ICTP research school of Mekn{\`e}s in
May 2012. To appear in the Proceedings of the conference "Quantum topology"
organized by Chelyabinsk State University in July 2014 (Vestnik ChelGU
A combinatorial definition of the Theta-invariant from Heegaard diagrams
The invariant is an invariant of rational homology 3-spheres
equipped with a combing over the complement of a point. It is related to
the Casson-Walker invariant by the formula
, where is an invariant of combings
that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a
combinatorial formula for the -invariant in terms of Heegaard diagrams,
equipped with decorations that define combings, from the definition of
as an algebraic intersection in a configuration space. In this article, we
prove that this formula defines an invariant of pairs without referring
to configuration spaces, and we prove that this invariant is the sum of and for integral homology spheres, by proving surgery
formulae both for the combinatorial invariant and for .Comment: 63 page
A combinatorial definition of the Theta-invariant from Heegaard diagrams
International audienceThe invariant is an invariant of rational homology 3-spheres equipped with a combing over the complement of a point. It is related to the Casson-Walker invariant by the formula , where is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a combinatorial formula for the -invariant in terms of Heegaard diagrams, equipped with decorations that define combings, from the definition of as an algebraic intersection in a configuration space. In this article, we prove that this formula defines an invariant of pairs without referring to configuration spaces, and we prove that this invariant is the sum of and for integral homology spheres, by proving surgery formulae both for the combinatorial invariant and for
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