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Champs radiaux sur une stratification analytique. (Radial fields on an analytic stratification). (French) Zbl 0727.57026
Travaux en Cours, 39. Paris: Hermann. ix, 183 p. FF 230.00 (1991).
This memoir, containing 6 sections and 4 appendices, deals with Whitney stratifications which are real semi-analytic and complex in certain cases. In the first part (§§1-3) certain vector fields (continuous, transversal, radial) tangent to the strata \(\{V_ i\}\) of the stratification are studied. In particular, for a radial field which has isolated zeros the conservation property of the indices at each zero belonging to \(V_ i\) with respect to the radial prolongation is proved. In the second part (§4), preradial fields are defined in the complex analytic case. This family of vector fields is larger than the family of radial fields, but it has all the useful properties. The author shows that to each preradial field associated to a complex analytic stratification a (noncanonical) radial field can be associated. The radial vector fields are used in order to formulate and prove a generalization of the Hopf theorem when the manifold is no longer smooth (§6). Also, a converse statement giving a simple condition in order that an outward vector field should be tangentially extensible to the strata, without zeros, is established. In order to prove these theorems, one needs some technical results on the neighborhoods of an analytic subset W of M, stratified by \(\{V_ i\}\), which are discussed in §5. Thus, the existence of a base of neighborhoods of W in M, called semi- geodesic tubes, is established. Some techniques and results of this work can be found in previous papers of the author [Acta Math. 91, 189–244 (1954; Zbl 0057.38102); “Classes caractéristiques d’un sous-ensemble analytique d’une variété analytique (multigraphié), Lille (1964–65), et repris dans Publ. U. E. R., Math. Pures Appl. 11 (1988); C. R. Acad. Sci., Paris, Sér. I 303, 239–241 (1986; Zbl 0604.58006) and ibid. 303, 307–309 (1986; Zbl 0604.57020)], but many of them appear here in an ameliorated and more detailed form. This memoir is highly recommended for those who want to learn fruitful tools used in the study of stratifications.

57R25 Vector fields, frame fields in differential topology
58A35 Stratified sets
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32C18 Topology of analytic spaces
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces