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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 astratification satisfying a strong (real arc-analytic with respect to allvariables and analytic with respect to the parameter space) trivializationproperty along each stratum. We call such a trivialization arc-wise analyticand we show that it can be constructed under the classical Zariskialgebro-geometric equisingularity assumptions. Using a slightly strongerversion of Zariski equisingularity, we show the existence of Whitney'sstratified fibration, satisfying the conditions (b) of Whitney and (w) ofVerdier. Our construction is based on Puiseux with parameter theorem and ageneralization of Whitney interpolation. For algebraic sets our constructiongives a global stratification.  We also give several applications of arc-wise analytic trivialization, mainlyto the stratification theory and the equisingularity of analytic set andfunction germs. In the real algebraic case, for an algebraic family ofprojective varieties, we show that Zariski equisingularity implies localtriviality of the weight filtration.

Arcwise Analytic Stratification, Whitney Fibering Conjecture and Zariski Equisingularity