Symplectic Instanton Homology: naturality, and maps from cobordisms

TL;DR

This paper establishes the naturality of symplectic instanton homology and defines cobordism maps, enabling new representations and geometric interpretations within the theory.

Contribution

It proves naturality of symplectic instanton homology and constructs cobordism maps, providing new tools for understanding 3- and 4-manifold invariants.

Findings

Symplectic instanton homology is shown to be natural.

Cobordism maps are defined within this framework.

New representations of the mapping class group and fundamental group are constructed.

Abstract

We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions.