Bordered Heegaard Floer Homology and four-manifolds with corners

Abstract

The Heegaard Floer hat invariant is defined on closed 3-manifolds, with a related invariant for 4-dimensional cobordisms, forming a 3+1 topological quantum field theory. Bordered Heegaard Floer homology generalizes this invariant to parametrized Riemann surfaces and to cobordisms between them, yielding a 2+1 TQFT. We discuss an approach to synthesizing these two theories to form a 2+1+1 TQFT, by defining Heegaard Floer invariants for Lefschetz fibrations with corners.