## Algebraic & Geometric Topology

Michael Farber

#### Abstract

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 $ℝ3$. We also prove results about higher moments of Betti numbers.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 155-171.

Dates
Accepted: 18 December 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796810

Digital Object Identifier
doi:10.2140/agt.2008.8.155

Mathematical Reviews number (MathSciNet)
MR2377280

Zentralblatt MATH identifier
1135.55009

#### Citation

Farber, Michael. Topology of random linkages. Algebr. Geom. Topol. 8 (2008), no. 1, 155--171. doi:10.2140/agt.2008.8.155. https://projecteuclid.org/euclid.agt/1513796810

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