# Variation formulas for an extended Gompf invariant

## TL;DR

This paper extends Gompf's homotopy invariant for oriented 2-plane fields in 3-manifolds to include manifolds with boundary and analyzes its behavior under surgeries, establishing it as a degree two finite type invariant.

## Contribution

It introduces an extension of Gompf's invariant applicable to all compact oriented 3-manifolds with boundary and studies its variations under Lagrangian-preserving surgeries.

## Key findings

- Extended Gompf invariant defined for manifolds with boundary.
- Invariant exhibits degree two behavior in finite type theory.
- Variation formulas under surgeries established.

## Abstract

In 1998, R. Gompf defined a homotopy invariant $\theta_G$ of oriented 2-plane fields in 3-manifolds. This invariant is defined for oriented 2-plane fields $\xi$ in a closed oriented 3-manifold $M$ when the first Chern class $c_1(\xi)$ is a torsion element of $H^2(M;\mathbb{Z})$. In this article, we define an extension of the Gompf invariant for all compact oriented 3-manifolds with boundary and we study its iterated variations under Lagrangian-preserving surgeries. It follows that the extended Gompf invariant is a degree two invariant with respect to a suitable finite type invariant theory.