Computable, obstructed Morse homology for clean intersections

TL;DR

This paper introduces a new computational approach for Morse homology in manifolds with clean, non-transverse intersections by constructing semi-global Kuranishi structures, bridging the gap between efficiency and generality.

Contribution

We develop minimal semi-global Kuranishi structures that generalize obstruction bundle gluing, enabling computable Morse homology for non-transverse intersections.

Findings

Constructed semi-global Kuranishi structures for Morse trajectories

Established that iterated gluing equals simultaneous gluing

Enhanced computational methods for Morse homology in complex 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.