Geodesic
On the product of cocycles in a~polyhedral complex
Izvestiya. Mathematics , Tome 81 (2017) no. 2, pp. 329-358

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2017_81_2_a4,
     author = {B. Ya. Kazarnovskii},
     title = {On the product of cocycles in a~polyhedral complex},
     journal = {Izvestiya. Mathematics },
     pages = {329--358},
     publisher = {mathdoc},
     volume = {81},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_2_a4/}
}
TY  - JOUR
AU  - B. Ya. Kazarnovskii
TI  - On the product of cocycles in a~polyhedral complex
JO  - Izvestiya. Mathematics 
PY  - 2017
SP  - 329
EP  - 358
VL  - 81
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2017_81_2_a4/
LA  - en
ID  - IM2_2017_81_2_a4
ER  - 
%0 Journal Article
%A B. Ya. Kazarnovskii
%T On the product of cocycles in a~polyhedral complex
%J Izvestiya. Mathematics 
%D 2017
%P 329-358
%V 81
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2017_81_2_a4/
%G en
%F IM2_2017_81_2_a4
B. Ya. Kazarnovskii. On the product of cocycles in a~polyhedral complex. Izvestiya. Mathematics , Tome 81 (2017) no. 2, pp. 329-358. http://geodesic.mathdoc.fr/item/IM2_2017_81_2_a4/