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

Abstract

We prove that the mixed isomorphisms of Daemi, Fukaya and Lipyanskiy type between framed instanton homology and the Lagrangian Floer homology of a Heegaard splitting intertwine the integral instanton cobordism maps, oriented by almost-complex structures, with signed symplectic cobordism maps, up to chain homotopy and one overall sign for each cobordism. This holds for every connected cobordism between connected three-manifolds carrying a framed arc and an $\mathrm{S}\mathrm{O}(3)$ bundle trivialized along the ends and the arc. Where the integral framed instanton homology is nonzero, the symplectic maps are signed counts of holomorphic triangles, with the Lagrangian correspondence of a compression body as one boundary condition for one- and three-handles. At three-manifolds whose integral framed instanton homology vanishes, the complexes are contractible and the maps are zero. The orientations are transported from the gauge theory through the mixed anti-self-duality and holomorphic curve equation. Consequently these maps form a projective functor on homology, independent of the handle decomposition up to sign. The unsigned analysis is that of the comparison modulo two; the new input is that two degenerations of the mixed equation, the collapse of a compression body and excision along a torus, preserve orientations. Where this homology is nonzero, we also prove that the sign by which the orientations change when a sphere is attached to a strip is constant under small perturbations and invariant under reversing the splitting. The relation with homology-oriented integral instanton maps is left open.