On the regular stratifications and conormal structure of subanalytic sets.

*(English)*Zbl 0585.32006The main theorem of this paper says that every subanalytic set admits a (b\({}^*)\)-regular stratification (see the definition below). The authors deal with the problem in an extremely interesting way. They give three different proofs of the main theorem, comparing the two methods of approach to subanalytic sets (with and without resolution of singularities) and using, in the third proof, the methods of conormal geometry that they developed.

An example is given showing that Whitney (b)-regularity need not always imply \((b^*)\)-regularity. This is connected with the natural question whether, for closed subanalytic sets, (b) implies \((b^*)\), as it is for (w) and \((w^*)\) or (r) and \((r^*)\) (Verdier, Navarro Aznar, Trotman).

The paper is very explanatory, as it shows the relations between different poblems, pointing out the connections between the regularity of stratifications and the density of Morse functions on stratified sets and the role of subanalytic sets in the microlocal study of PDE.

Definition of \((b^*)\)-regularity: The pair (X,Y) of disjoint \(C^ 1\)- submanifolds of \({\mathbb{R}}^ n\) is said to be \((b^*)\)-regular at \(y\in \bar X\) if for all k, \(0\leq k<n-\dim Y\), there exists an open dense subset \({\mathcal U}\) of the Grassmannian manifold \({\mathcal W}=\{Z\in {\mathcal G}_{n-k}({\mathbb{R}}^ n): Z\supset T_ yY\},\) such that for every W which is a \(C^ 1\)-submanifold of \({\mathbb{R}}^ n\) transversal to X near Y, containing Y and whose tangent space \(T_ yW\) belongs to \({\mathcal U}\), the pair (X\(\cap W,Y)\) satisfies Whitney (b)-condition at y.

An example is given showing that Whitney (b)-regularity need not always imply \((b^*)\)-regularity. This is connected with the natural question whether, for closed subanalytic sets, (b) implies \((b^*)\), as it is for (w) and \((w^*)\) or (r) and \((r^*)\) (Verdier, Navarro Aznar, Trotman).

The paper is very explanatory, as it shows the relations between different poblems, pointing out the connections between the regularity of stratifications and the density of Morse functions on stratified sets and the role of subanalytic sets in the microlocal study of PDE.

Definition of \((b^*)\)-regularity: The pair (X,Y) of disjoint \(C^ 1\)- submanifolds of \({\mathbb{R}}^ n\) is said to be \((b^*)\)-regular at \(y\in \bar X\) if for all k, \(0\leq k<n-\dim Y\), there exists an open dense subset \({\mathcal U}\) of the Grassmannian manifold \({\mathcal W}=\{Z\in {\mathcal G}_{n-k}({\mathbb{R}}^ n): Z\supset T_ yY\},\) such that for every W which is a \(C^ 1\)-submanifold of \({\mathbb{R}}^ n\) transversal to X near Y, containing Y and whose tangent space \(T_ yW\) belongs to \({\mathcal U}\), the pair (X\(\cap W,Y)\) satisfies Whitney (b)-condition at y.

Reviewer: Z.Denkowska

##### MSC:

32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |