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Algebraic approximation of manifolds and spaces

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Algebraic approximation of manifolds and spaces SÉMINAIRE N. BOURBAKI A. TOGNOLI Algebraic approximation ofmanifolds and spaces Séminaire N. Bourbaki, 1979-1980, exp. no 548, p. 73-94. <> © Association des collaborateurs de Nicolas Bourbaki, 1979-1980, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. implique l’accord avec les conditions générales d’utilisa- tion ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques ALGEBRAIC APPROXIMATION OF MANIFOLDS AND SPACES by A. TOGNOLI 73 Seminaire BOURBAKI 32e annee, 1979/80, nO 548 Novembre 1979 Introduction We can state the two following informal problems : 00 I - given a compact C manifold M is it possible to induce on M a real algebraic structure M a such that the geometry of M a can be described by algebraic elements ? (for example, such that any a £ H (M ,S~,) can be represented by an algebraic cycle). II - characterize the topological spaces that are homeomorphic to a singular real algebraic variety. The main progress in the study of these problems can be summarized as follows : Seifert studied problem I in the case of the complete intersections (1936), Whitney showed (in the analytic case) that problem I could be treated also in the general case (1936), Nash gave a partial solution of the problem using the Whitney methods in the real algebraic case (1952), Wallace demonstrated that any compact manifold, which is a 00 boundary, has an algebraic structure (1957), we proved that any compact C manifold has algebraic structure (1973). Now we have also some information about particular algebraic structures, in which a part

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