A refinement of Morse–Novikov cohomology on manifolds with boundary and the cohomology of the space of solutions of kerΔ𝜃

In this paper, we consider the Morse–Novikov coboundary operator Formula: see text where Formula: see text, and which is given by Formula: see text, and for which the associated Morse–Novikov cohomology groups are denoted by Formula: see text. The ultimate objective of this paper is to uncover more geometric and topological insights about Formula: see text when the boundary of Formula: see text is nonvanishing. To begin, we employ a different approach to prove that the concrete realizations of the Morse–Novikov cohomology groups of Formula: see text intersect at the origin, and then we use the long exact sequence and cohomological algebra as powerful tools to decompose the absolute and relative Morse–Novikov cohomologies into interior and boundary portions, respectively. On the other hand, we use a novel reasoning based on cohomological algebra to determine the cohomology of Formula: see text, and as a result the absolute Morse–Novikov cohomology of Formula: see text can be characterized in terms of two successive degrees in just one degree of the cohomology of Formula: see text. Consequently, the cohomology of Formula: see text can be recovered from the interior and boundary sections.