Symplectic instanton homology: naturality, and maps from cobordisms

Abstract

We prove that Manolescu and Woodward’s symplectic instanton homology, and its twisted versions, are natural; and definemaps associated to four dimensional cobordisms within this theory.

This allows one to define representations of the mapping class group, the fundamental group and the first cohomology group with ${\mathbb{Z}}_2$ coefficients of a 3-manifold. We also provide a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions.