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Preprint Number 1144

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1144. David Trotman and Guillaume Valette
On the local geometry of definably stratified sets

Submission date: 18 January 2017


(This paper will appear in the proceedings of a 2015 conference in memory of Murray Marshall, to be published in the A.M.S. Contemporary Mathematics series.)

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Pawlucki's theorem for arbitrary o-minimal structures, replacing Whitney (b)-regularity by a quantified version, and we prove related results concerning normal cones and continuity of the density. We analyse two counterexamples to the extension of Pawlucki's theorem to definable sets in general o-minimal structures, and to several other statements valid for subanalytic sets. In particular we give the first example of a Whitney (b)-regular definably stratified set for which the density is not continuous along a stratum.

Mathematics Subject Classification: 32S15 (Primary), 03C10, 58A35 (Secondary)

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Full text arXiv 1701.05087: pdf, ps.

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