Cobordism maps in Khovanov homology and singular instanton homology I

TL;DR

This paper constructs a rigorous cobordism map linking Khovanov homology and singular instanton homology, establishing a direct correspondence between their functorial structures via chain maps.

Contribution

It defines a filtered chain map on the instanton cube complex that recovers cobordism maps in both homologies, bridging the gap between the theories.

Findings

Defined a cobordism map as a filtered chain map

Proved the map recovers Khovanov and instanton cobordism maps

Established a rigorous link between the two homology theories

Abstract

Khovanov homology and singular instanton Floer homology are both functorial with respect to link cobordisms. Although the two homology groups are related by a spectral sequence, direct correspondence between the cobordism maps has not been rigorously established. In this paper, we define a cobordism map on the instanton cube complex as a filtered chain map, and prove that it recovers the cobordism maps both in Khovanov homology and singular instanton theory. In a sequel paper, we further extend this cobordism map to immersed cobordisms.