Algebraic cycles and algebraic models of smooth manifolds

By Tognoli's theorem, any smooth compact manifold M has an algebraic model, that is, there exists a nonsingular real algebraic set X diffeomorphic to M. In fact, one can find an uncountable family of pairwise nonisomorphic al gebraic models of M, assuming that M has a positive dimension. In the prese nt paper we are concerned with the group of homology classes on X (with int eger coefficients modulo 2) that are represented by d-dimensional algebraic subsets of X. We investigate how this group varies as X runs through the c lass of all algebraic models of M.