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Russian Mathematical Surveys, 2005, Volume 60, Issue 4, Pages 615–644
DOI: https://doi.org/10.1070/RM2005v060n04ABEH003671
(Mi rm1444)
 

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

Computation of characteristic classes of a manifold from a triangulation of it

A. A. Gaifullin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
Abstract: This paper is devoted to the well-known problem of computing the Stiefel–Whitney classes and the Pontryagin classes of a manifold from a given triangulation of the manifold. In 1940 Whitney found local combinatorial formulae for the Stiefel–Whitney classes. The first combinatorial formula for the first rational Pontryagin class was found by Gabrielov, Gel'fand, and Losik in 1975. Since then, different authors have constructed several different formulae for the rational characteristic classes of a triangulated manifold, but none of these formulae provides an algorithm that computes the characteristic cycle solely from a triangulation of the manifold. In this paper a new local combinatorial formula recently found by the author for the first Pontryagin class is described; it provides the desired algorithm. This result uses a solution of the following problem: construct a function $f$ on the set of isomorphism classes of three-dimensional PL-spheres such that for any combinatorial manifold the chain obtained by taking each simplex of codimension four with coefficient equal to the value of the function on the link of the simplex is a cycle.
Received: 06.06.2005
Russian version:
Uspekhi Matematicheskikh Nauk, 2005, Volume 60, Issue 4(364), Pages 37–66
DOI: https://doi.org/10.4213/rm1444
Bibliographic databases:
Document Type: Article
UDC: 515.164.3
MSC: Primary 57R20, 55U10; Secondary 57Q15, 14C17, 57Q20
Language: English
Original paper language: Russian
Citation: A. A. Gaifullin, “Computation of characteristic classes of a manifold from a triangulation of it”, Uspekhi Mat. Nauk, 60:4(364) (2005), 37–66; Russian Math. Surveys, 60:4 (2005), 615–644
Citation in format AMSBIB
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\paper Computation of characteristic classes of a~manifold from a triangulation of~it
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 4(364)
\pages 37--66
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\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 4
\pages 615--644
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Linking options:
  • https://www.mathnet.ru/eng/rm1444
  • https://doi.org/10.1070/RM2005v060n04ABEH003671
  • https://www.mathnet.ru/eng/rm/v60/i4/p37
  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:1065
    Russian version PDF:499
    English version PDF:38
    References:62
    First page:2
     
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