630 research outputs found
A proof of Reidemeister-Singer's theorem by Cerf's methods
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 solving 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
On the codimension-one foliation theorem of W. Thurston
This article has been withdrawn due to a mistake which is explained in
version 2.Comment: 1 pag
A Note on the Chas-Sullivan product
We give a finite dimensional approach to the Chas-Sullivan product on the
free loop space of a manifold, orientable or not
A proof of Morse's theorem about the cancellation of critical points
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
A Morse complex on manifolds with boundary
Given a compact smooth manifold 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 with integer
coefficients. Our approach simplifies other methods which have been discussed
in more specific geometric settings
Essential curves in handlebodies and topological contractions
If is a compact set, a {\it topological contraction} is a self-embedding
such that the intersection of the successive images , ,
consists of one point. In dimension 3, we prove that there are smooth
topological contractions of the handlebodies of genus whose image is
essential. Our proof is based on an easy criterion for a simple curve to be
essential in a handlebody
On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows
We show that if a sequence of Hamiltonian flows has a limit, and if the
generating Hamiltonians of the sequence have a limit, then this limit is
uniquely determned by the limiting flow. This answers a question by Y.G.
Oh.Comment: 11 page
Regularization of Gamma_1-structures in dimension 3
For -structures on 3-manifolds, we give a very simple proof of
Thurston's regularization theorem, first proved in \cite{thurston}, without
using Mather's 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
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