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Betti numbers of random manifolds
Author(s) -
Farber Michael,
Kappeler Thomas
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700709
Subject(s) - betti number , mathematics , infinity , combinatorics , class (philosophy) , polygon (computer graphics) , discrete mathematics , mathematical analysis , computer science , artificial intelligence , telecommunications , frame (networking)
Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R 3 . We also prove results about higher moments of Betti numbers. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)