Homology Classes Represented by Real Algebraic Hypersurfaces

摘要

Let M be a (C sup infinity symbol) manifold. A nonsingular affine real algebraic variety (i.e., up to biregular isomorphism, an algebraic subset of R sup n) diffeomorphic to M is called an algebraic model of M. The celebrated theorem of Tognoli says that each compact (C sup infinity symbol) manifold admits an algebraic model. In the paper, the authors study the problem of constructing, for a given compact (C sup infinity symbol) manifold M of dimension d, an algebraic model X of M whose some homology classes in (H sub (d-1))(X,Z/2) cannot be represented by algebraic hypersurfaces of X.