ALGEBRAIC CYCLES AND APPROXIMATION THEOREMS IN REAL ALGEBRAIC-GEOMETRY

Let M be a compact C(infinity) manifold. A theorem of Nash-Tognoli ass erts that M has an algebraic model, that is, M is diffeomorphic to a n onsingular real algebraic set X. Let H(alg)k (X, Z/2) denote the subgr oup of H(k)(X, Z/2) of the cohomology classes determined by algebraic cycles of codimension k on X. Assuming that M is connected, orientable and dim M greater-than-or-equal-to 5, we prove in this paper that a s ubgroup G of H2(M, Z/2) is isomorphic to H(alg)2(X, Z/2) for some alge braic model X of M if and only if W2(TM) is in G and each element of G is of the form w2(xi) for some real vector bundle xi over M, where w2 Stands for the second Stiefel-Whitney class. A result of this type wa s previously known for subgroups G of H-1 (M, Z/2).