---
abstract: 'We define a twisted version of Manolescu and Woodward’s Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward’s Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.'
address: |
    Laboratoire de Mathématiques Jean Leray\
    Université de Nantes\
    2, rue de la Houssinière - BP 92208 F-44322 Nantes Cedex 3\
    France
author:
- Guillem Cazassus
bibliography:
- 'biblio.bib'
title: 'Symplectic Instanton Homology: twisting, connected sums, and Dehn surgery'
---

Introduction {#sec:intro}
============

Instanton homology groups are graded abelian groups associated to integral homology 3-spheres introduced by Floer in [@Floerinst]. They form natural receptacles for analogs of Donaldson invariants for 4-manifolds with boundary, and allow one to compute Donaldson invariants by cutting a closed 4-manifold along 3-manifolds. Indeed, they satisfy axioms of a (3+1)-topological quantum field theory in the sense of Atiyah (with the exception that the groups are not defined for all 3-manifolds).

Likewise, Atiyah suggested in [@Atiyahfloer] the following alternative procedure for defining these invariants by cutting 3-manifolds along surfaces, called the Atiyah-Floer conjecture. By stretching a 3-manifold $Y$ along a Heegaard surface $\Sigma$ dividing $Y$ into two handlebodies $H_0$ and $H_1$, he suggested that these invariants can 