We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.We define the weight filtration of the homology of a real algebraic variety by first addressing the case of smooth noncompact varieties. As in Deligne's definition [5] of the weight filtration for complex varieties, given a smooth variety X we consider a good compactification, a smooth compactification X of X such that D = X \ X is a divisor with normal crossings. Whereas Deligne's construction can be interpreted in terms of the action of a torus (S 1 ) N , we use the action of a discrete torus (S 0 ) N to define a filtration of the chains of a semialgebraic compactification of X associated to the divisor D. The resulting filtered chain complex is functorial for pairs (X, X) as above, and it behaves nicely for a blowup with a smooth center that has normal crossings with D.We apply a result of Guillén and Navarro Aznar [6, Theorem (2.3.6)] to show that our filtered complex is independent of the good compactification of X (up to quasi-isomorphism) and to extend our definition to a functorial filtered complex, the weight complex, that is defined for all varieties and enjoys a generalized blowup property (Theorem 7.1). For compact varieties the weight complex agrees with our previous definition [9] for Borel-Moore homology.We work with homology rather than cohomology to take advantage of the topology of semialgebraic chains [9, Appendix]. We denote by H k (X) the kth classical homology group of X, with compact supports and coefficients in Z 2 , the integers modulo 2. The vector space H k (X) is dual to H k (X), the classical kth cohomology group with closed supports. On the other hand, let H BM k (X) denote the kth Borel-Moore homology group of X (i.e. homology with closed supports) with coefficients in Z 2 . Then H BM k (X) is dual to H k c (X), the kth cohomology group with compact supports. Our work owes much to the foundational paper [6] of Guillén and Navarro Aznar. In particular we have been influenced by the viewpoint of section 5 of that paper, on the theory of motives. Using Guillén and Navarro Aznar's extension theorems, Totaro [13] observed that there is a functorial weight filtration for the cohomology with compact supports of a real analytic variety with a given compactification. In [9] we developed this theory in detail for real algebraic varieties, working with Borel-Moore homology. Our task was simplified by the strong additivity property of Borel-Moore homology (or compactly supported cohomology) [9, Theorem 1.

Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x 0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x 0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x 0 onto the unit sphere has finite length. IntroductionLet f be a real analytic function on an open set U ⊂ R n and let ∇f be its gradient in the Euclidean metric. We shall study the trajectories of ∇f , i.e. the maximal curves x(t) satisfyingIn the sixties Lojasiewicz [Lo2] (see also [Lo4]) proved the following result.Lojasiewicz's Theorem. If x(t) has a limit point x 0 ∈ U , i.e. x(t ν ) → x 0 for some sequence t ν → β, then the length of x(t) is finite; moreover, β = ∞. Therefore x(t) → x 0 as t → ∞.Note that ∇f (x 0 ) = 0, since otherwise we could extend x(t) through x 0 . The purpose of this paper is to prove the following statement, called "the gradient conjecture of R. Thom" (see [Th]Gradient Conjecture. Suppose that x(t) → x 0 . Then x(t) has a tangent at x 0 , that is the limit of secants lim t→∞ x(t) − x 0 |x(t) − x 0 | exists.

The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then βi(X) = βi(X \ Y ) + βi(Y ).We define an invariant of the Grothendieck ring K 0 (V R ) of real algebraic varieties, the virtual Poincaré polynomial β(X, t), which is a ring homomorphism K 0 (V R ) → Z [t]. For X nonsingular and compact, β(X, t) is the classical Poincaré polynomial for cohomology with Z 2

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.