ABSTRACT. Given a compact smooth hypersurface M in Rn+1, we construct a family {Xk}, k = 1,2,..., of nonsingular real algebraic subsets of Rn+1 such that each Xk is isotopic to M but, for k / I, Xk and X¡ are not birationally equivalent.

Abstract. Let M be a compact connected orientable C°° submanifold of R" with 2 dim M + 1 < n . Let G be a subgroup of H2(M, Z) such that the quotient group H2(M,Z) has no torsion. Then M can be approximated in R" by a nonsingular algebraic subset X such that H^.(X, Z) is isomorphic to G. Here H£_. (X,Z) denotes the subgroup of H2(X,Z) generated by the cohomology classes determined by the complex algebraic hypersurfaces in a complexification of X .