, Volume 125, Issue 1, pp 75-92

Homology of planar polygon spaces

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In this paper, we study topology of the variety of closed planar n-gons with given side lengths \(l_1, \dots, l_n\) . The moduli space \(M_\ell\) where \(\ell =(l_1, \dots, l_n)\) , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces \(M_\ell\) as functions of the length vector \(\ell=(l_1, \dots, l_n)\) . We also find sharp upper bounds on the sum of Betti numbers of \(M_\ell\) depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.