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Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism. (English) Zbl 0547.14019

Let V be a closed smooth manifold. It is well-known that there exists a non singular real algebraic set (shortly an algebraic manifold) V’ diffeomorphic to V (Tognoli). It was a longstanding problem if there would exist such a V’ such that \(H_*(V',Z_ 2)=H_*^{(a)}(V')=^{def}the\quad subgroup\) generated by the fundamental classes of the algebraic subvarieties of V’. A positive answer should be very useful, for example, in many questions about the topology of real algebraic sets. On the contrary it is proved here that the answer is, in general, negative: in fact for every \(d\geq 11\) it is constructed such a manifold V of dimension d, such that for any homeomorphism \(h:V\to V'\) between V and an algebraic manifold V’, \(h_*(z)\) does not belong to \(H^{(a)}_{d-2}(V')\), where z is a suitable element of \(H_{d-2}(V)\). Analogous counterexamples are produced for the unoriented differentiable bordism groups of V.

MSC:

14Pxx Real algebraic and real-analytic geometry
57R95 Realizing cycles by submanifolds
57R19 Algebraic topology on manifolds and differential topology
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References:

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