Algebraic Cycles and Approximation Theorems in Real Algebraic Geometry

Abstract

Abstract. Let Af be a compact C°° manifold. A theorem of Nash-Tognoli asserts that M has an algebraic model, that is, M is diffeomorphic to a nonsingular real algebraic set X . Let FV^AfX, Z/2) denote the subgroup of Hk(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 > 5 , we prove in this paper that a subgroup G of H2(M, Z/2) is isomorphic to H^ (X, Z/2) for some algebraic model X of M if and only if w2(TM) is in G and each element of G is of the form W2K) for some real vector bundle £ over M , where w2 stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups G of HX(M, Z/2).

Stiefel—Whitney classes for coherent real analytic sheaves

“…for example [4,Proposition 1.4]). If X is compact of dimension at most 5, then W (X) = H 2 (X, Z/2).…”

Cycles on Nash algebraic models of smooth manifolds

“…Since then a lot of effort has been devoted to the construction of algebraic models with certain additional desirable properties; see [1,2,3,4,5,6,7,8,9,18]. In the recent papers [13,14,15,19], a relaxed concept of algebraic model is used, which is closer in spirit to Nash's original paper [20].…”

“…If X 0 is the union of some connected components of X, we denote by H alg k (X 0 , Z/2) the subgroup of H k (X, Z/2) consisting of the homology classes represented by k-dimensional algebraic subsets of X contained in X 0 . The behavior of the groups H alg k (X, Z/2) (or their counterparts in cohomology) as X runs through the class of all algebraic models of a given compact smooth manifold M is investigated in [7,8,9,18].…”

Algebraic obstructions and a complete solution of a rational retraction problem

“…Such a question may be subdivided in two complementary problems: 1-total obstruction problem) find smooth compact manifolds M and homology classes α ∈ H * (M, Z/2Z) such that, in every algebraic model M of M , α is not represented by any real algebraic subset of M ; 2-constructive problem) given a subgroup G of H k (M, Z/2Z) for some integer k, find an algebraic model M such that G corresponds to the subgroup H alg k (M , Z/2Z) of H k (M , Z/2Z) generated by the homology classes represented by all k-dimensional real algebraic subsets of M . The latter problem has been investigated by Bochnak and Kucharz in [11] and [12] (see also section 11.3 of [8]). With regard to the first problem Benedetti and Dedò [5] proved the following theorem which, in the final analysis, even today remains the unique result regarding algebraic obstructions up to homeomorphism.…”