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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 5, Pages 13–66 (Mi izv502)  

This article is cited in 8 scientific papers (total in 8 papers)

Local formulae for combinatorial Pontryagin classes

A. A. Gaifullin


Abstract: Let $p(|K|)$ be the characteristic class of a combinatorial manifold $K$ given by a polynomial $p$ in the rational Pontryagin classes of $K$. We prove that for any polynomial $p$ there is a function taking each combinatorial manifold $K$ to a cycle $z_p(K)$ in its rational simplicial chains such that: 1) the Poincaré dual of $z_p(K)$ represents the cohomology class $p(|K|)$; 2) the coefficient of each simplex $\Delta$ in the cycle $z_p(K)$ is determined solely by the combinatorial type of $\operatorname{link}\Delta$. We explicitly describe all such functions for the first Pontryagin class. We obtain estimates for the denominators of the coefficients of the simplices in the cycles $z_p(K)$.

DOI: https://doi.org/10.4213/im502

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English version:
Izvestiya: Mathematics, 2004, 68:5, 861–910

Bibliographic databases:

UDC: 515.164.3
MSC: Primary 57Q15; Secondary 57R20, 55R40, 55R60
Received: 08.06.2004

Citation: A. A. Gaifullin, “Local formulae for combinatorial Pontryagin classes”, Izv. RAN. Ser. Mat., 68:5 (2004), 13–66; Izv. Math., 68:5 (2004), 861–910

Citation in format AMSBIB
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\paper Local formulae for combinatorial Pontryagin classes
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\pages 13--66
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Gaifullin, “Computation of characteristic classes of a manifold from a triangulation of it”, Russian Math. Surveys, 60:4 (2005), 615–644  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. A. Gaifullin, “Explicit construction of manifolds realising prescribed homology classes”, Russian Math. Surveys, 62:6 (2007), 1199–1201  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. A. Gaifullin, “The construction of combinatorial manifolds with prescribed sets of links of vertices”, Izv. Math., 72:5 (2008), 845–899  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. A. Gaifullin, “The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds”, Proc. Steklov Inst. Math., 263 (2008), 38–56  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    5. Alexander A. Gaifullin, “Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class”, Proc. Steklov Inst. Math., 268 (2010), 70–86  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. D. A. Gorodkov, “A minimal triangulation of the quaternionic projective plane”, Russian Math. Surveys, 71:6 (2016), 1140–1142  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. A. Gaifullin, D. A. Gorodkov, “An explicit local combinatorial formula for the first Pontryagin class”, Russian Math. Surveys, 74:6 (2019), 1120–1122  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Gorodkov D., “a 15-Vertex Triangulation of the Quaternionic Projective Plane”, Discret. Comput. Geom., 62:2 (2019), 348–373  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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