Invariance of Heegaard Floer maps for cobordisms with corners

Establish whether the maps on bordered Heegaard Floer invariants associated to 4-dimensional cobordisms with corners are well-defined invariants of the cobordism (independent of choices such as handle decompositions), i.e., prove that for any cobordism with corners X the induced map F_X on the appropriate bordered Floer modules is invariant up to homotopy under diffeomorphisms rel boundary.

Background

The paper constructs maps associated to complements of slice disks and more general cobordisms with corners using Heegaard Floer techniques and handle decompositions. While the resulting glued maps are shown to be independent of choices after pairing, the authors note that, in general, it has not been proved that the maps defined prior to gluing are invariants of the underlying cobordisms with corners.

Clarifying this invariance would strengthen the functoriality picture for bordered Floer homology and underpin applications that distinguish 4-manifolds and embedded surfaces by their induced maps, without relying on subsequent gluing to closed cobordisms.