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 $M$. It only depends on the Spin$^c$-structure represented by the combing. When this Euler class is a torsion element of $H^2(M;Z)$, we say that the combing is a torsion combing. Gompf introduced a $Q$-valued invariant $\theta_G$ of torsion combings of closed 3-manifolds that distinguishes all combings that represent a given Spin$^c$-structure. This invariant provides a grading of the Heegaard Floer homology $\hat{HF}$ for manifolds equipped with torsion Spin$^c$-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 $p_1$ for combings of manifolds bounded by $S^2$. We show that the $\Theta$-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 $(\frac14p_1 + 6 \lambda)$ where $\lambda$ 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 $\Theta$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $\lambda$ by the formula $\Theta(M,X)=6\lambda(M)+p_1(X)/4$, where $p_1$ is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a combinatorial formula for the $\Theta$-invariant in terms of Heegaard diagrams, equipped with decorations that define combings, from the definition of $\Theta$ as an algebraic intersection in a configuration space. In this article, we prove that this formula defines an invariant of pairs $(M,X)$ without referring to configuration spaces, and we prove that this invariant is the sum of $6 \lambda(M)$ and $p_1(X)/4$ for integral homology spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$.Comment: 63 page

Provision of club goods: cost sharing and selection of a provider

This paper characterizes optimal mechanisms facilitating the cost sharing and the selection of a provider for a club good. These mechanisms are allocatively and Pareto efficient. However, it appears that transfers occur even when the good is not provided. This result is due to the weakening of the incentive notion to Bayesian-Nash equilibrium and to the balanced budget condition. This phenomena disappears if the setting is perfectly symmetric.

A combinatorial definition of the Theta-invariant from Heegaard diagrams

International audienceThe invariant $\Theta$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $\lambda$ by the formula $\Theta(M,X)=6\lambda(M)+p_1(X)/4$, where $p_1$ is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a combinatorial formula for the $\Theta$-invariant in terms of Heegaard diagrams, equipped with decorations that define combings, from the definition of $\Theta$ as an algebraic intersection in a configuration space. In this article, we prove that this formula defines an invariant of pairs $(M,X)$ without referring to configuration spaces, and we prove that this invariant is the sum of $6 \lambda(M)$ and $p_1(X)/4$ for integral homology spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$