Local formulae for combinatorial Pontrjagin classes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2004
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| _version_ | 1866915026125717504 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | By p(|K|) denote the characteristic class of a combinatorial manifold K given by the polynomial p in Pontrjagin classes of K. We prove that for any polynomial p there exists a function taking each combinatorial manifold K to a rational simplicial cycle z(K) such that: (1) the Poincare dual of z(K) represents the cohomology class p(|K|); (2) a coefficient of each simplex in the cycle z(K) is determined only by the combinatorial type of the link of this simplex. We also prove that if a function z satisfies the condition (2), then this function automatically satisfies the condition (1) for some polynomial p. We describe explicitly all such functions z for the first Pontrjagin class. We obtain estimates for denominators of coefficients of simplices in the cycles z(K). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0407035 |
| institution | arXiv |
| publishDate | 2004 |
| record_format | arxiv |
| spellingShingle | Local formulae for combinatorial Pontrjagin classes Gaifullin, Alexander A. Algebraic Topology Combinatorics By p(|K|) denote the characteristic class of a combinatorial manifold K given by the polynomial p in Pontrjagin classes of K. We prove that for any polynomial p there exists a function taking each combinatorial manifold K to a rational simplicial cycle z(K) such that: (1) the Poincare dual of z(K) represents the cohomology class p(|K|); (2) a coefficient of each simplex in the cycle z(K) is determined only by the combinatorial type of the link of this simplex. We also prove that if a function z satisfies the condition (2), then this function automatically satisfies the condition (1) for some polynomial p. We describe explicitly all such functions z for the first Pontrjagin class. We obtain estimates for denominators of coefficients of simplices in the cycles z(K). |
| title | Local formulae for combinatorial Pontrjagin classes |
| topic | Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/math/0407035 |