Abstract
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.
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References
Hausmann, J.-Cl.: Sur la topologie des bras articulés. In: Algebraic Topology, Poznan. Lecture Notes, vol. 1474, pp. 146 – 159. Springer, Heidelberg (1989)
Hausmann J.-Cl. and Knutson A. (1998). Cohomology rings of polygon spaces. Ann. Inst. Fourier (Grenoble) 48: 281–321
Hausmann J.-C. and Rodriguez E. (2004). The space of clouds in an Euclidean space. Exp. Math. 13: 31–47
Kamiyama Y., Tezuka M. and Toma T. (1998). Homology of the configuration spaces of quasi-equilateral polygon linkages. Trans. AMS 350: 4869–4896
Kamiyama Y. and Tezuka M. (1999). Topology and geometry of equilateral polygon linkages in the Euclidean plane. Q. J. Math. 50: 463–470
Kapovich M. and Millson J.L. (1995). On the moduli space of polygons in the Euclidean plane. J. Differ. Geometry 42: 133–164
Kapovich M. and Millson J.L. (1996). The symplectic geometry of polygons in Euclidean space. J. Differ. Geometry 44: 479–513
Klyachko, A.A.: Spatial polygons and stable configurations of points in the projective line. Algebraic geometry and its applications. Aspects Math., E25, pp 67–84. Vieweg, Braunschweig (1994)
Milgram R.J. and Trinkle J.C. (2004). The geometry of configuration spaces for closed chains in two and three dimensions. Homol. Homot. Appl. 6: 237–267
Milnor J. (1969). Morse theory. Princeton University Press, Princeton
Milnor J. (1966). Lectures on the h-cobordism theorem. Princeton University Press, Princeton
Thurston W. and Weeks J. (1984). The mathematics of three-dimensional manifolds. Sci. Am. 251: 94–106
Vardi I. (1991). Computational recreations in mathematics. Addison-Wesley, Redwood City, CA
Walker, K.: Configuration spaces of linkages. Undergraduate thesis, Princeton (1985)
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Farber, M., Schütz, D. Homology of planar polygon spaces. Geom Dedicata 125, 75–92 (2007). https://doi.org/10.1007/s10711-007-9139-7
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DOI: https://doi.org/10.1007/s10711-007-9139-7