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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.
Received: 31.07.2015
Citation:
B. Ya. Kazarnovskii, “On the product of cocycles in a polyhedral complex”, Izv. Math., 81:2 (2017), 329–358
Linking options:
https://www.mathnet.ru/eng/im8435https://doi.org/10.1070/IM8435 https://www.mathnet.ru/eng/im/v81/i2/p97
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Abstract page: | 410 | Russian version PDF: | 50 | English version PDF: | 14 | References: | 60 | First page: | 26 |
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