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Weakly Whitney stratified sets. (English) Zbl 0942.58010
Bruce, J. W. (ed.) et al., Real and complex singularities. Proceedings of the 5th workshop, São Carlos, Brazil, July 27-31, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 412, 1-15 (2000).
The authors define weakly Whitney stratified spaces and prove that these spaces form an intermediate class between classical Whitney stratified spaces and the locally radial $$(C)$$-regular spaces which they had studied in previous papers.
First they introduce a metric condition $$(\delta)$$ satisfied by every Whitney stratification. Weakly Whitney stratifications are those stratifications which satisfy both condition $$(a)$$ and condition $$(\delta)$$. So they possess many properties of Whitney stratifications. An interesting consequence is the possibility of topologically modeling biological forms such as shells and horns. They also give examples of real algebraic varieties in $$\mathbb{R}^3$$ and show that one may have conditions $$(a)$$ and $$(\delta)$$ without $$(b)$$, and that one may have $$(\delta)$$ without $$(a)$$. Furthermore, they prove good topological properties of weakly Whitney stratifications: the class is invariant by transverse intersection and each stratification is $$(C)$$-regular for standard control functions. Finally, they note that weakly Whitney stratified sets in Riemannian manifolds are locally radial.
For the entire collection see [Zbl 0922.00019].

##### MSC:
 58A35 Stratified sets 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)