Homology of Planar Polygon Spaces
نویسندگان
چکیده
In this paper we study topology of the variety of closed planar n-gons with given side lengths l1, . . . , ln. The moduli space M` where ` = (l1, . . . , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M` as functions of the length vector ` = (l1, . . . , ln). We also find sharp upper bounds on the sum of Betti numbers of M` 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.
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