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 Uspekhi Mat. Nauk, 2005, Volume 60, Issue 4(364), Pages 37–66 (Mi umn1444)

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

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.

DOI: https://doi.org/10.4213/rm1444

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English version:
Russian Mathematical Surveys, 2005, 60:4, 615–644

Bibliographic databases:

UDC: 515.164.3
MSC: Primary 57R20, 55U10; Secondary 57Q15, 14C17, 57Q20

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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• http://mi.mathnet.ru/eng/umn1444
• https://doi.org/10.4213/rm1444
• http://mi.mathnet.ru/eng/umn/v60/i4/p37

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This publication is cited in the following articles:
1. A. A. Gaifullin, “The construction of combinatorial manifolds with prescribed sets of links of vertices”, Izv. Math., 72:5 (2008), 845–899
2. Alexander A. Gaifullin, “Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class”, Proc. Steklov Inst. Math., 268 (2010), 70–86
3. Li Yu, “Localizable invariants of combinatorial manifolds and Euler characteristic”, Arch. Math, 2014
4. J. Math. Sci. (N. Y.), 224:2 (2017), 304–327
5. Govc D., Marzantowicz W., Pavesic P., “How Many Simplices Are Needed to Triangulate a Grassmannian?”, Topol. Methods Nonlinear Anal., 56:2 (2020), 501–518
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