Computable, obstructed Morse homology for clean intersections

Abstract

In this paper, we develop a method to compute the Morse homology of a manifold when descending manifolds and ascending manifolds intersect cleanly, but not necessarily transversely. While obstruction bundle gluing defined by Hutchings and Taubes is a computable tool to handle non-transverse intersections, it has only been developed for specific cases. In contrast, most virtual techniques apply to general cases but lack computational efficiency. To address this, we construct minimal semi-global Kuranishi structures for the moduli spaces of Morse trajectories, which generalize obstruction bundle gluing while maintaining its computability feature. Through this construction, we obtain iterated gluing equals simultaneous gluing.