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Bi-Lipschitz equisingularity. (English) Zbl 1231.32018

Manoel, M. (ed.) et al., Real and complex singularities. Selected papers of the 10th international workshop held at the University of São Paulo (ICMC-USP), São Carlos, Brazil, July 27–August 2, 2008. Cambridge: Cambridge University Press (ISBN 978-0-521-16969-1/pbk). London Mathematical Society Lecture Note Series 380, 338-349 (2010).
This paper presents a survey on equisingularity of analytic varieties (or semialgebraic or subanalytic sets) focussing on bi-Lipschitz equisingularity. The author starts with a review of Whitney and Verdier stratifications in the real and complex settings. Next, he discusses o-minimal structures, a more general category of sets which includes the semialgebraic sets and mappings. These sets also admit a Whitney stratification. It is well known that given an analytic variety (or semialgebraic set or subanalytic set) with some Whitney stratification, then the stratified set is locally trivial along each stratum by the first isotopy theorem of Thom-Mather. The existence of a bi-Lipschitz stratification, as introduced by Mostowski, and the discussion of similarities and differences of this theory and Whitney stratification theory are presented. Finally, the paper reviews Valette’s triangulation theorem and its consequences.
For the entire collection see [Zbl 1202.00100].

MSC:

32S15 Equisingularity (topological and analytic)
14P10 Semialgebraic sets and related spaces
14P15 Real-analytic and semi-analytic sets
32B15 Analytic subsets of affine space
32B20 Semi-analytic sets, subanalytic sets, and generalizations
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
58A35 Stratified sets
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