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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 2, Pages 97–128 (Mi izv8435)  

On the product of cocycles in a polyhedral complex

B. Ya. Kazarnovskii

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: We construct an algorithm for multiplying cochains in a polyhedral complex. It depends on the choice of a linear functional on the ambient space. The cocycles form a subring in the ring of cochains, the coboundaries form an ideal in the ring of cocycles, and the quotient ring is the cohomology ring. The multiplication algorithm depends on the geometry of the cells of the complex. For simplicial complexes (the simplest geometry of cells), it reduces to the well-known Čech algorithm. Our algorithm is of geometric origin. For example, it applies in the calculation of mixed volumes of polyhedra and the construction of stable intersections of tropical varieties. In geometry it is customary to multiply cocycles with values in the exterior algebra of the ambient space. Therefore we assume that the ring of values is supercommutative.

Keywords: product of cocycles, polyhedral complex, polyhedron, tropical variety.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
This work was supported by the Russian Science Foundation (grant no. 14-50-00150) and carried out in the Institute of Information Transmission Problems (Kharkevich Institute) of the Russian Academy of Sciences.


DOI: https://doi.org/10.4213/im8435

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English version:
Izvestiya: Mathematics, 2017, 81:2, 329–358

Bibliographic databases:

UDC: 514+515.14
MSC: 55N45, 52A39, 52B70, 14M25
Received: 31.07.2015

Citation: B. Ya. Kazarnovskii, “On the product of cocycles in a polyhedral complex”, Izv. RAN. Ser. Mat., 81:2 (2017), 97–128; Izv. Math., 81:2 (2017), 329–358

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