Cobordism maps and Atiyah–Floer isomorphisms for framed instanton homology I

Abstract

We prove that isomorphisms of the type constructed by Daemi, Fukaya and Lipyanskiy, between framed instanton homology and the Lagrangian Floer homology of a Heegaard splitting, intertwine the cobordism maps of Kronheimer and Mrowka with maps defined by holomorphic curves alone, over the field of two elements. This holds for every connected cobordism between connected three-manifolds carrying a framed arc between its ends and an $\mathrm{S}\mathrm{O}(3)$ bundle trivialized along the ends and the arc. The symplectic maps count holomorphic triangles in moduli spaces of flat connections on surfaces, with the correspondence of a compression body as boundary condition for one- and three-handles. Consequently these maps are independent of the handle decomposition and of the Heegaard data, and form a functor isomorphic to framed instanton homology; where both are defined, our isomorphisms agree with those of Daemi, Fukaya and Lipyanskiy modulo two. The proof rests on two analytic results for the mixed anti-self-duality and holomorphic curve equation: an adiabatic limit in which a compression body collapses next to a holomorphic strip and becomes the Lagrangian boundary condition of its flat connections, and an excision theorem along a torus.