Local formulae for combinatorial Pontryagin classes

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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 zp(K) in its rational simplicial chains such that: 1) the Poincaré dual of zp(K) represents the cohomology class p(|K|); 2) the coefficient of each simplex Δ in the cycle z p(K) is determined solely by the combinatorial type of link Δ. 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 Zp(K).

Original languageEnglish
Pages (from-to)861-910
Number of pages50
JournalIzvestiya Mathematics
Volume68
Issue number5
DOIs
Publication statusPublished - Sep 2004
Externally publishedYes

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