Homology and cohomology classes represented by real algebraic sets

Let X be a compact real algebraic variety. Denote by HalgkX,Z /2 the sub group of HalgkX,Z /2 generated by the homology classes represented by k-dimensional subvarieties of X. If X is nonsingular we define the algebraic cohomology groups HalgkX,Z /2 via Poincare duality.;We adopt a scheme-theoretic point of view and discuss how the groups Hkalg are related to the theory of algebraic cycles, especially to algebraic equivalence. We show that the subgroup Algk of cohomology classes determined by algebraic cycles algebraically equivalent to zero must satisfy quite restrictive conditions. Given a manifold M, these conditions characterize the spherical cohomology classes of M that belong to Algk for some algebraic structure on M.;We also consider the problem of finding algebraic models of a given manifold with prescribed Halg1 . We obtain several results in some particular situations and a very general result in the case of an n-dimensional torus.;The group Halg1 is also studied for real hyperelliptic surfaces and real projective ruled surfaces. Numerical equivalence of algebraic cycles is used to show that Halg1 is a proper subgroup of H1 , for a nonorientable hyperelliptic surface. We follow the minimal model program to give the list of all possible groups Halg1 for all the topological types of real projective ruled surfaces.