## Topology of real algebraic sets.(English)Zbl 0808.14045

Mathematical Sciences Research Institute Publications. 25. New York: Springer-Verlag. x, 249 p. (1992).
The book is devoted to the following fundamental problem of real algebraic geometry: to classify topologically real algebraic sets, to give a topological characterization of all topological spaces which are homeomorphic to real algebraic sets. Main results in this direction achieved at the moment are presented in a comprehensive and self- contained setting. Namely, a complete topological characterization of nonsingular algebraic sets, of algebraic sets with only isolated singularities and of algebraic sets of dimension $$\leq 3$$ is given, as well as the affirmative solution to the Nash conjecture that every smooth manifold in $$\mathbb{R}^ n$$ is smoothly isotopic to a close nonsingular algebraic set. A detailed introduction to the methods used in this topic is done. In particular, the theory of resolution towers is developed, which is a far generalization of resolution of singularities adapted to the problem.

### MSC:

 14P25 Topology of real algebraic varieties 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 32C05 Real-analytic manifolds, real-analytic spaces 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57N80 Stratifications in topological manifolds 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 58A35 Stratified sets