Topology of Real Algebraic Varieties

Abstract

In the first section, we prove some combinatorial topological properties of real algebraic sets; the simplest and most important of these properties is the fact that, for every semi-algebraic triangulation of a bounded algebraic set of dimension d and every (d - 1)-simplex σ of such a triangulation, the number of d-simplices of the triangulation having σ as a face is even. In the second section, we use this property and an appropriate stratification to prove that, for every point a of an algebraic set V, the local Euler-Poincaré characteristic χ(V, V \ a) is odd; this result gives a necessary combinatorial condition for a polyhedron to be homeomorphic to a real algebraic set. In Section 3 we define the fundamental ℤ/2-homology class of a real algebraic variety. This leads to the concept of algebraic homology groups of a real algebraic variety, consisting of the homology classes represented by algebraic subsets. These groups, which are basic invariants, will be used in Chap. 12 and 13. We construct examples of nonsingular algebraic sets whose homology is not totally algebraic. In Section 4, we use the Borel-Moore fundamental classes to prove that an injective regular mapping from a nonsingular irreducible algebraic set to itself is surjective. The analogous result in complex algebraic geometry (without the assumption of nonsingularity) is well known, but the methods of proof are completely different. Section 5 contains an upper bound for the sum of the Betti numbers of an algebraic set. Section 6 is devoted to algebraic curves in the real projective plane. We prove Harnack’s theorem concerning the maximum number of connected components of a nonsingular curve of given degree and some results concerning the first part of Hilbert’s 16^th problem (without proving the crucial Rokhlin congruence, Theorem 11.6.4).

For algebraic subsets of ℝⁿ, the homology H _* (resp. (H _* ^BM) used in this chapter is the usual singular homology (resp. the Borel-Moore homology for locally compact spaces). In order to extend the results to the case of an arbitrary real closed field, we need a homology theory for semi-algebraic sets, with the properties of the usual singular homology. We explain how to construct such a theory in the appendix to this chapter. This construction is only sketched. Nevertheless, we hope to convince the reader that this homology theory behaves like singular homology theory in those situations we are interested in. For this reason, in Sections 2 to 5 we use a homology theory for semi-algebraic sets over a real closed field, which will be explained only in the appendix (Section 7). The appendix also serves as a reference for the Borel-Moore homology used in Section 4; the subtleties of this theory are not needed in the semi-algebraic framework.

Throughout this chapter, R denotes a real closed field.