For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromovs h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of c
odimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametri
In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer.
Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for so
lving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M>2. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.
We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried
by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurstons 1976 regularization theorem without using Mathers homology equivalence.
This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from cite{GrLaPo}, cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which f
its well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy function reduces to the type of embedding of one-dimensional attractors and repellers of a given Morse-Smale diffeomorphism on a closed 3-manifold.
Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or relative to the
boundary) homology of $M$ with integer coefficients. Our approach simplifies other methods which have been discussed in more specific geometric settings.
The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambien
t manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.
For $Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurstons regularization theorem, first proved in cite{thurston}, without using Mathers homology equivalence. Moreover, in the co-orientable case, the resulting foliation can be
chosen of a precise kind, namely an open book foliation modified by suspension. There is also a model in the non co-orientable case.
