A Fibration Theorem and Chas-Sullivan product for Morse-Novikov Homology with differential graded coefficients

TL;DR

This paper extends Morse-Novikov Homology with differential graded coefficients to include a fibration-based fibration theorem and a Chas-Sullivan product, providing new algebraic structures for these homologies.

Contribution

It introduces a Morse-Novikov complex with differential graded coefficients and establishes a fibration theorem and a Chas-Sullivan-like product in this context.

Findings

Constructed a Morse-Novikov complex with differential graded coefficients.

Proved existence of a Morse-Novikov model for Novikov completion of homology.

Established a Chas-Sullivan-like product under certain fibration conditions.

Abstract

Given an oriented, closed and connected manifold X and a nonzero cohomology u $\in$ H 1 (X, R), we extend the constructions of Morse Homology with differential graded coefficients of [BDHO25] and of the Chas-Sullivan product described on this model in [Rie24] to Morse-Novikov Homology with differential graded coefficients. More precisely, we will construct a Morse-Novikov complex with differential graded coefficients and prove that, given a fibration E $\pi$ $\rightarrow$ X such that $\pi$ * u = 0 $\in$ H 1 (E, R) and the fiber is locally path-connected, there exists a Morse-Novikov model for a ''Novikov completion'' H * (E, u) of the singular homology of E. Moreover, we will prove that if the fibration is endowed with a morphism of fibrations on its fibers, then there exists a Chas-Sullivan-like product on H * (E, u) that can be described within this model.