ABSTRACT. Using the work of Guillén and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to a filtered quasi-isomorphism, and induces on Borel— Moore homology with ℤ2 coefficients an analog of the weight filtration for complex algebraic varieties.

The weight complex can be represented by a geometrically defined filtration on the complex of semialgebraic chains. To show this we define the weight complex for Nash manifolds and, more generally, for arc-symmetric sets, and we adapt to Nash manifolds the theorem of Mikhalkin that two compact connected smooth manifolds of the same dimension can be connected by a sequence of smooth blowups and blowdowns.

The weight complex is acyclic for smooth blowups and additive for closed inclusions. As a corollary we obtain a new construction of the virtual Betti numbers, which are additive invariants of real algebraic varieties, and we show their invariance by a large class of mappings that includes regular homeomorphisms and Nash diffeomorphisms.