Algebraic models of smooth manifolds (Q1262915)

Let X be a nonsingular real affine algebraic variety with \(\dim (X)=d\) and let \(H^ k_{alg}(X,{\mathbb{Z}}/2)\) be the image under the Poincaré duality isomorphism of the subgroup of \(H_{d-k}(X,{\mathbb{Z}}/2)\) consisting of homology classes represented by Zariski closed \((d-k)\)- dimensional algebraic subvarieties of X. The groups \(H^ k_{alg}(X,{\mathbb{Z}}/2)\) are very important invariants of X but still the knowledge of their behavior seems rather limited, in spite of several related results scattered in the literature. The main problem considered in the paper under review is the following. Does every compact smooth manifold admit an algebraic model with prescribed \(H^ 1_{alg} ?\) A classical result of Tognoli states that each compact smooth manifold M admits an algebraic model, i.e. is diffeomorphic to an X as above. The question is whether for any subgroup G of \(H^ 1(M,{\mathbb{Z}}/2)\) there are an algebraic model X of M and a diffeomorphism \(\phi: X\to M\) such that \(\phi^*G=H^ 1_{alg}(X,{\mathbb{Z}}/2)\). The authors answer this question in case \(\dim(M)\geq 3\), M connected, by proving that if M is orientable, then for any \(G\subset H^ 1(M,{\mathbb{Z}}/2)\) there exist X and \(\phi\) as above, while if M is nonorientable, the existence of X and \(\phi\) as above is equivalent to \(G\ni w_ 1(M)\), the first Stiefel-Whitney class of M. In case of surfaces the authors provide a partial result in terms of two numerical invariants related to \(H^ 1_{alg}(X,{\mathbb{Z}}/2)\) which, as they show, also determine the projective module \(K_ 0({\mathcal R}(X))\) of the ring \({\mathcal R}(X)\) of real regular functions on X. Among the applications the authors prove a factoriality result for \({\mathcal R}(X)\), when M is orientable with \(\dim(M)\geq 2\), and obtain information about the subset of \(C^{\infty}(X,S^ 1)\) consisting of smooth mappings into the unit circle approximable by regular ones.