Elimination of extremal index zeroes from generic paths of closed 1-forms

TL;DR

This paper proves that generic paths of Morse closed 1-forms on a manifold can be deformed to eliminate centers throughout the path, extending known results about individual forms to continuous families.

Contribution

It introduces a one-parameter analogue of the known result that non-vanishing cohomology classes admit Morse forms without centers, showing such forms can be connected through paths avoiding centers.

Findings

Paths of closed 1-forms can be modified to eliminate centers entirely.

The result applies to generic paths with fixed endpoints having no centers.

It extends the understanding of the topology of Morse forms in a cohomology class.

Abstract

Let $\alpha$ be a Morse closed $1$-form of a smooth $n$-dimensional manifold $M$. The zeroes of $\alpha$ of index $0$ or $n$ are called \emph{centers}. It is known that every non-vanishing de Rham cohomology class $u$ contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path $ (\alpha_t)_{t\in [0,1]}$ of closed $1$-forms in a fixed class $u\neq 0$ such that $\alpha_0, \alpha_1$ have no centers, can be modified relatively to its extremities to another such path $ (\beta_t)_{t\in [0,1]}$ having no center at all.