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On metric properties of stratified sets. (English) Zbl 1033.58005

Topological properties of Thom-Whitney stratified spaces have been much studied by several mathematicians. The first author [Lect. Notes Math. 1462, 42–62 (1991; Zbl 0733.58003)] showed that after replacing Whitney’s condition (b) by a weaker condition, called (C)-regularity, one still obtains most of the topological properties of Whitney stratified spaces, e.g., existence of a Thom-Mather structure as an abstract stratified set defined by a system of control data, invariance by transverse intersection and continuous controlled extension of stratified vector fields.
In the paper under review, the authors study the class of locally radial stratified spaces which are (C)-regular for standard control functions. A locally compact regular stratified space is locally homeomorphic to a cone on a regular stratified space. The volume is locally finite and in fact equivalent to that of a ball of the same dimension. Stokes’ theorem for certain regular stratified spaces is proved. Finally, the weakly Whitney stratified spaces are described. This class is intermediate between the classical Whitney stratified spaces and locally radial (C)-regular spaces.

MSC:

58A35 Stratified sets
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
58C35 Integration on manifolds; measures on manifolds
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)

Citations:

Zbl 0733.58003
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