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

Carlos Moraga Ferrandiz (1, 2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Graduate School of Mathematical Sciences

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.

Mots clés: Morse-Novikov theory; Cerf theory; closed 1-forms; pseudo-gradients; pseudo-isotopy; s-cobordism

Domaines: Topologie géométrique [math.GT]

Carlos Moraga Ferrandiz. Elimination of extremal index zeroes from generic paths of closed 1-forms. Mathematische Zeitschrift, 2014, 25 pp. ⟨10.1007/s00209-014-1332-4⟩. ⟨hal-00803918v3⟩