On homology classes represented by real algebraic varieties

Abstract

Introduction. Let X be a compact real algebraic variety (essentially, a compact real algebraic set, see Section 1). Denote by H k (X,Z/2) the subgroup of Hk(X,Z/2) generated by the homology classes represented by Zariski closed k-dimensional subvarieties of X. If X is nonsingular and d = dimX, let H alg(X,Z/2) be the subgroup of H(X,Z/2) consisting of all the cohomology classes that are sent via the Poincare duality isomorphism H(X,Z/2)→ Hd−k(X,Z/2) into H d−k(X,Z/2). In this short paper we survey certain results concerning the groups H k and H k alg, and their applications. Section 1 contains the precise definitions of these groups (based on a construction of the fundamental homology class of a compact real algebraic variety) and theorems establishing their functorial properties and relating them to the StiefelWhitney classes of algebraic vector bundles. With the exception of Theorem 1.7 (ii), all the results (modulo minor modifications) come from the classical source [15]. In Section 2 we adopt a scheme-theoretic point of view and discuss how the groups H k are related to the theory of algebraic cycles, especially rational and algebraic equivalence. In particular, we give a purely algebraic geometric description of H k . Our main references are [15, 17, 25, 28]. In Section 3 we study how the groups H alg(X,Z/2), for k = 1, 2, vary as X runs through the class of nonsingular real algebraic varieties diffeomorphic to a given closed C∞ manifold. We rely mostly on [9, 11, 37], but the reader may also wish to consult