Absracts of papers
Vanishing homology:
In this paper we introduce a new homology theory devoted to the study
of families such as semi-algebraic or subanalytic families and in general to any family
definable in an o-minimal structure (such as Denjoy-Carleman definable or ln - exp
definable sets). The idea is to study the cycles which are vanishing when we approach a
special fiber. This also enables us to derive local metric invariants for germs of definable
sets. We prove that the homology groups are finitely generated.
Article in pdf
Bi-Lipschitz sufficiency of jets:
We give some theorems of bi-Lipschitz or C1 sufficiency of jets
which are expressed by means of transversality with respect to
some strata of a stratification satisfying the (L) condition of
T. Mostowski. This enables us to prove that the number of metric
types of intersection of smooth transversals to a stratum of a
(a) regular stratification of a subanalytic set is finite.
Article in pdf
Multiplicity mod 2 as a semi-algebraic bi-Lipschitz invariant:
We study the multiplicity mod 2 of real algebraic hypersurfaces. We prove that under some assumptions
on the singularity it is preserved through a semi-algebraic bi-Lipschitz homeomorphism of Sn.
In a first part we find a part of the tangent cone enclosing the multiplicity mod 2 and prove that it is an equivariant subset of Sn.
Studying equivariant submanifolds of Sn we are able to conclude about its invariance through semi-algebraic bi-Lipschitz
homeomorphisms under an extra assumption on the tangent cone.
Article in pdf
Volume, Whitney conditions and Lelong number:
This paper studies the variation of the volume of a subanalytic family of
sets. More precisely we are interested in the variation of the density. We prove that the
density is continuous along a stratum of a Whitney subanalytic stratification and locally
lipschitzian when the stratification satisfies the Kuo-Verdier condition. This problem
had been studied by G. Comte in [C].
Article in pdf
[C] G. Comte, Equisingularite reelle : nombres de Lelong et images polaires. (French) [Real equisingularity:
Lelong numbers and polar images] Ann. Sci. Ecole Norm. Sup. (4) 33 (2000), no. 6,
757–788.
On metric types that are definable in an o-minimal structure:
In this paper we study the metric spaces that are definable in a polynomially
bounded o-minimal structure. We prove that the family of metric spaces definable in a
given polynomially bounded o-minimal structure is characterized by the valuation field
k of the structure. In the last section we prove that the cardinality of this family is
that of k. In particular these two results answer a conjecture given in [SS] about the
countability of the metric types of analytic germs. The proof is a mixture of geometry
and model theory.
Article in pdf
[SS] L. Siebenmann, D. Sullivan, On complexes that are Lipschitz manifolds. Geometric topology (Proc.
Georgia Topology Conf., Athens, Ga., 1977), pp. 503–525, Academic Press, New York-London,
1979.
The link of the germ of a semi-algebraic metric space:
In this paper we investigate the metric properties of semi-algebraic germs.
More precisely we introduce a counterpart to the notion of link for semi-algebraic metric
spaces, which is often used to study the topology. We prove that it totally determines the
metric type of the germ. We give a nice consequence for semi-algebraically bi-Lipschitz
homeomorphic semi-algebraic germs.
Lipschitz triangulations:
In this paper we introduce a new tool called "Lipschitz
triangulations", which gives combinatorially all information about the
metric type. We show the existence of such triangulations for semialgebraic
sets. As a consequence we obtain a bi-Lipschitz version of
Hardt's theorem. Hardt's theorem states that, given a family definable
in an o-minimal structure, there exists (generically) a trivialization
which is definable in this o-minimal structure. We show that, for a
polynomially bounded o-minimal structure, there exists such an isotopy
which is bi-Lipschitz as well.
Article in pdf
Volume and multiplicities of real analytic sets:
We give criteria of finite determinacy for the volume and
multiplicities. Given an analytic set described by v=0, we
prove that the log-analytic expansion of the volume of the
intersection of the set by a "little ball" is determined by that
of the set defined by the Taylor expansion of v at a certain
order if the mapping v has an isolated singularity at the
origin. We also compare the cardinals of finite projections
restricted to such a set.
A bilipschitz version of Hardt's theorem:
In this note we give a sketch of the proof of a theorem which is a
bilipschitz version of Hardt's theorem [H]. Given a family
definable in an o-minimal structure Hardt's theorem states the
existence (for generic parameters) of a trivialization which is
definable in the o-minimal structure. We show that, for a
polynomially bounded o-minimal structure, there exists such an
isotopy which is bilipschitz.
The proof is inspired by [BCR1] and involves the construction
of "lipschitz triangulations''
which are defined in this note. The complete proof of existence will appear in [V].
[BCR1] J. Bochnak, M. Coste and M.-F. Roy, Geometrie algebrique reelle, Ergebnisse
der Mathematik und ihrer Grenzgebiete (3), vol. 12, Springer-Verlag, Berlin, 1987.
[H] R. M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent.
Math. 38 (1976/77), 207-217.
[V] G. Valette, Lipschitz triangulations. Illinois J. Math. 49 (2005), issue 3, 953979.
Lipschitz stratifications and generic wings (with Dwi Juniati and David Trotman):
The paper shows that, for subanalytic stratifications, Lipschitz equisingularity as defined by Mostowski is preserved
after intersection with generic wings, that is, L-regularity implies L*-regularity. This was one of the conditions
required of a good equisingularity notion by Teissier in his foundational 1974 Arcata paper.
Previous authors have shown that Lipschitz equisingularity is generic, implies bilipschitz triviality, and hence topological triviality, and implies equimultiplicity.