The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{\infty }$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

For a germ (X,0) of normal complex space of dimension n + 1 with an isolated singularity at 0 and a germ f: (X,0) → (ℂ,0) of holomorphic function with df(x) ≤ 0 for x ≤ 0, the fibre-integrals     $s\mapsto {\int }_{f=s}\varrho {\omega }^{\text{'}}\bigwedge \overline{{\omega }^{\text{'}\text{'}}},\varrho \in C_c^{\infty }\left(X\right),{\omega }^{\text{'}},{\omega }^{\text{'}\text{'}}\in {\Omega }_X^n$, are $C^{\infty }$ on ℂ* and have an asymptotic expansion at 0. Even when f is singular, it may happen that all these fibre-integrals are $C^{\infty }$. We study such maps and build a family of examples where also fibre-integrals for ${\omega }^{\text{'}},{\omega }^{\text{'}\text{'}}\in {⍹}_X$, the Grothendieck sheaf, are $C^{\infty }$.

Dans notre article [6] nous avons construit, pour une classe assez large de germes de fonctions holomorphes $f:({ℂ}^{n+1},0)\to (ℂ,0)$ à lieu singulier $S:=\{df=0\}$ de dimension 1 des invariants analytiques qui généralisent le réseau de Brieskorn d’un germe à singularité isolée. Dans cet article nous montrons que les résultats que nous avions obtenus s’étendent àtous les germes à lieu singulier de dimension 1 sans autre restriction. Ces invariants, essentiellement donnés par des (a,b)-modules géométriques, (objet qui est une abstraction...

We prove the “End Curve Theorem,” which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\Sigma$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function $(X,o)\to (ℂ,0)$ whose zero set intersects $\Sigma$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” $(X,o)$ is described by giving an explicit set of equations describing...

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.

Assume that $\Gamma$ is a connected negative definite plumbing graph, and that the associated plumbed 3-manifold $M$ is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg–Witten invariant of $M$. The first one is the constant term of a ‘multivariable Hilbert polynomial’, it reflects in a conceptual way the structure of the graph $\Gamma$, and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The second...

We express the Lyubeznik numbers of the local ring of a complex isolated singularity in terms of Betti numbers of the associated real link.

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

Caustics of geometrical optics are understood as special types of Lagrangian singularities. In the compact case, they have remarkable topological properties, expressed in particular by the Chekanov relation. We show how this relation may be experimentally checked on an example of biperiodic caustics produced by the deflection of the light by a nematic liquid crystal layer. Moreover the physical laws may impose a geometrical constraint, when the system is invariant by some group of symmetries. We...

L’objet de cet article est de démontrer un théorème «  à la Thom-Sebastiani  » pour les développements asymptotiques des intégrales-fibres des fonctions du type $f\oplus g:(x,y)\to f\left(x\right)+g\left(y\right)$ sur $({ℂ}^p\times {ℂ}^q,(0,0))$ en terme des développements asymptotiques des intégrales-fibres associées aux germes holomorphes $f:({ℂ}^p,0)\to (ℂ,0)$ et $g:({ℂ}^q,0)\to (ℂ,0)$. Ceci se ramène à calculer les développements asymptotiques d’une convolution $\Phi *\Psi$ à partir des développements asymptotiques de $\Phi$ et $\Psi$ modulo les termes non singuliers.Pour obtenir un résultat précis donnant la non nullité des termes...