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Analyticity of homology classes


Author: Alberto Tognoli
Journal: Proc. Amer. Math. Soc. 104 (1988), 920-922
MSC: Primary 57R95; Secondary 32C05
DOI: https://doi.org/10.1090/S0002-9939-1988-0964874-2
MathSciNet review: 964874
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Abstract: Let $ W$ be a real analytic manifold and $ \{ \alpha \} \in {H_p}(W,{Z_2})$. We shall say that $ \{ \alpha \} $ is analytic if there exists a compact analytic subset $ S$ of $ W$, such that: $ \{ \alpha \}=$   fundamental class of $S$$ \}$. The purpose of this short paper is to prove

Theorem 1. Let $ W$ be a paracompact real analytic manifold; then any homology class $ \{ \alpha \} \in {H_p}(W,{Z_2})$ is analytic.

We remember that a similar result does not hold in the real algebraic case (see [1]).


References [Enhancements On Off] (What's this?)

  • [1] R. Benedetti and M. Dedò, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math. 53 (1984), no. 2, 143–151. MR 766294
  • [2] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518
  • [3] A. Tognoli, Algebraic approximation of manifolds and spaces, Sem. Bourbaki 548 (1979-80).
  • [4] Riccardo Benedetti and Alberto Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), no. 1, 89–112 (English, with French summary). MR 560747
  • [5] Armand Borel and André Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461–513 (French). MR 0149503
  • [6] Aldo Andreotti and Per Holm, Quasianalytic and parametric spaces, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 13–97. MR 0589903
  • [7] A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 167–185. MR 0396571
  • [8] René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 0061823, https://doi.org/10.1007/BF02566923
  • [9] Alberto Tognoli, Algebraic geometry and Nash functions, Institutiones Mathematicae [Mathematical Methods], III, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 556239
  • [10] Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. An introduction to the fundamental theory; With the assistance of Gottfried Barthel; Translated from the German by Michael Bridgland. MR 716497

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0964874-2
Article copyright: © Copyright 1988 American Mathematical Society