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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.

Complex cycles on real algebraic models of a smooth manifold