N. Mnev, G. Sharygin, “On local combinatorial formulas for Chern classes of a triangulated circle bundle”, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Zap. Nauchn. Sem. POMI, 448, POMI, St. Petersburg, 2016, 201–235; J. Math. Sci. (N. Y.), 224:2 (2017), 304

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Zapiski Nauchnykh Seminarov POMI, 2016, Volume 448, Pages 201–235 (Mi znsl6312)

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

On local combinatorial formulas for Chern classes of a triangulated circle bundle

N. Mnev^ab, G. Sharygin^cd

^a St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia
^b Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia
^c Institute for Theoretical and Experimental Physics, Moscow, Russia
^d Moscow State University, Moscow, Russia

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Abstract: A principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in the combinatorial sense). We express rational local formulas for all powers of the first Chern class in terms of expectations of the parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of a triangulated circle bundle over a simplex, measuring the mixing by the triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deducing these formulas from Kontsevitch's cyclic invariant connection form on metric polygons.

Key words and phrases: circle bundle, Chern class, local formula.

Funding agency	Grant number
Russian Science Foundation	14-21-00035
Russian Foundation for Basic Research	14-01-00007
The main result of the paper (Theorem 4.1) was supported by the Russian Science Foundation grant 14-21-00035. G. Sharygin was additionally supported by the RFBR grant 14-01-00007.

Received: 16.08.2016

English version:
Journal of Mathematical Sciences (New York), 2017, Volume 224, Issue 2, Pages 304–327
DOI: https://doi.org/10.1007/s10958-017-3416-2

Bibliographic databases:

Document Type: Article

UDC: 515.145.2

Language: English

Citation: N. Mnev, G. Sharygin, “On local combinatorial formulas for Chern classes of a triangulated circle bundle”, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Zap. Nauchn. Sem. POMI, 448, POMI, St. Petersburg, 2016, 201–235; J. Math. Sci. (N. Y.), 224:2 (2017), 304–327

Citation in format AMSBIB

\Bibitem{MneSha16}

\by N.~Mnev, G.~Sharygin

\paper On local combinatorial formulas for Chern classes of a~triangulated circle bundle

\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXVII

\serial Zap. Nauchn. Sem. POMI

\yr 2016

\vol 448

\pages 201--235

\publ POMI

\publaddr St.~Petersburg

\mathnet{http://mi.mathnet.ru/znsl6312}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3576259}

\transl

\jour J. Math. Sci. (N. Y.)

\yr 2017

\vol 224

\issue 2

\pages 304--327

\crossref{https://doi.org/10.1007/s10958-017-3416-2}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85019686152}

Linking options:

https://www.mathnet.ru/eng/znsl6312

https://www.mathnet.ru/eng/znsl/v448/p201

This publication is cited in the following 8 articles:

G. Yu. Panina, “An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fibre bundle”, Sb. Math., 214:3 (2023), 429–443
N. E. Mnëv, “A Note on a Local Combinatorial Formula for the Euler Class of a PL Spherical Fiber Bundle”, J Math Sci, 261:5 (2022), 614
N. E. Mnëv, “A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXIII, Zap. nauchn. sem. POMI, 507, POMI, SPb., 2021, 35–58
N. Mnëv, “Minimal Triangulations of Circle Bundles, Circular Permutations, and the Binary Chern Cocycle”, J Math Sci, 247:5 (2020), 696
N. Mnëv, “Minimal triangulations of circle bundles, circular permutations, and the binary Chern cocycle”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 87–107
J. Gordon, G. Panina, “A combinatorial formula for monomials in Kontsevich's $\psi$ -classes”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXI, Zap. nauchn. sem. POMI, 485, POMI, SPb., 2019, 72–77
J. A. Gordon, G. Yu. Panina, “Diagonal complexes”, Izv. Math., 82:5 (2018), 861–879
J. Math. Sci. (N. Y.), 240:5 (2019), 551–555

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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References:	48

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