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.


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