Abstract. In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases.For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney's interpolation. For algebraic sets our construction gives a global stratification.We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.

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Let A ⊂ R n be a set-germ at 0 ∈ R n such that 0 ∈ A. We say thatWe study the problem of whether the dimension of the common direction set, dim(D(A) ∩ D(B)) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension.

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In this paper we study fundamental directional properties of sets under the assumption of condition (SSP ) (introduced in [3]). We show several transversality theorems in the singular case and an (SSP )-structure preserving theorem. As a geometric illustration, our transversality results are used to prove several facts concerning complex analytic varieties in 3.3. Also, using our results on sets with condition (SSP), we give a classification of spirals in the appendix 5.The (SSP )-property is most suitable for understanding transversality in the Lipschitz category. This property is shared by a large class of sets, in particular by subanalytic sets or by definable sets in an o-minimal structure.2010 Mathematics Subject Classification. Primary 14P15, 32B20, Secondary 57R45.