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

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 $u\ne 0$ 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.

arXiv commentMathematische Zeitschrift (2014) 25 pp

References in corpus (1)

• A proof of Morse's theorem about the cancellation of critical points

Cited by in corpus (1)

• Homoclinic bifurcation in Morse-Novikov theory, a doubling phenomenon