  In this paper we study topology of the variety of closed planar polygons with
given side lengths. We describe the Betti numbers of the moduli spaces as
functions of the length vector. We also find sharp upper bounds on the sum of
Betti numbers of the moduli space 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.
