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

Abstract

Let $α$ be a Morse closed $1$-form of a smooth $n$-dimensional manifold $M$. The zeroes of $α$ 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 $ (α_t)_{t\in [0,1]}$ of closed $1$-forms in a fixed class $u\neq 0$ such that $α_0, α_1$ have no centers, can be modified relatively to its extremities to another such path $ (β_t)_{t\in [0,1]}$ having no center at all.