Absracts of papers
A Lefschetz duality for intersection homology:
We prove a Lefschetz duality theorem for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed
to have a “collared neighborhood of their boundary”. Our duality does not need this assumption and is a generalization of the classical one.
Vector bundles and regulous maps:
Let X be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on X becomes algebraic after finitely many blowing ups.
Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic R -vector bundle on X are algebraic. We also derive that the Chern classes of
any pre-algebraic C -vector bundles and the Pontryagin classes of any pre-algebraic R -vector bundle are blow-C -algebraic.
We also provide several results on line bundles on X .
On bi-Lipschitz stability of families of functions:
We focus on the Lipschitz stability of families of functions. We introduce a stability notion, called fiberwise bi-Lipschitz equivalence,
which preserves the metric structure of the level surfaces of functions and show that it does not admit continuous moduli in the framework
of semialgebraic geometry. We trivialize semialgebraic families of Lipschitz functions by constructing triangulations of their generic fibers
which contain information about the metric structure of the sets.
De Rham Theorem for L∞ forms and homology on singular spaces:
We introduce smooth L∞ differential forms on a singular (semialgebraic) set X in Rn.
Roughly speaking, a smooth L∞ differential form is a certain class of equivalence of ``stratified forms",
that is, a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components
on the adjacent strata and bounded size (in the metric induced from Rn).
We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices,
i.e. continuous semialgebraic maps from the standard simplices into X. Singular cohomology of X is defined as the homology
of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural
isomorphism between the singular cohomology and the cohomology of smooth L∞ forms.
Article in pdf
L∞ cohomology is intersection cohomology:
Let X be any subanalytic compact pseudomanifold. We show a De Rham
theorem for L∞ forms. We prove that the cohomology of L∞ forms is isomorphic to
intersection cohomology in the maximal perversity.
C0and bi-Lipschitz K-equivalence:
In this paper we investigate the classification of mappings up to K-equivalence.
We give several results of this type. We study semialgebraic deformations up to semialgebraic
C0 K-equivalence and bi-Lipschitz K-equivalence. We give an algebraic criterion
for bi-Lipschitz K-triviality in terms of semi-integral closure (Theorem 3.5). We also
give a new proof of a result of Nishimura: we show that two germs of smooth mappings
f, g : Rn -> Rn, finitely determined with respect to K-equivalence are C0-K-equivalent if
and only if they have the same degree in absolute value.
Article in pdf
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 metric invariant:
We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove
that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved
by subanalytic bi-Lipschitz homeomorphisms of Rn. In the first part of the paper, we
find a subset of the tangent cone which determines the multiplicity mod 2 and prove
that this subset of Sn is preserved by the antipodal map. The study of such subsets of
Sn enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2
under some extra assumptions on the tangent cone. We also prove a real version of a
theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic
bi-Lipschitz homeomorphisms.
Article in pdf
Bi-Lipschitz trivial quasi-homogeneous stratifications:
We call a stratification quasi-homogeneous when it is invariant
by a certain class of action. The paper gives criteria for a
quasi-homogeneous stratification to fulfill the (w)-condition
of Kuo-Verdier or to be bi-Lipschitz trivial. We also give some
explicit conditions on the weights to ensure some stability of
the volume of quasi-homogeneous families of germs.
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, 953-979.
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.