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Topology and geometry of equilateral polygon linkages in the Euclidean plane. (English) Zbl 0961.57014

The paper under review is to give an elementary proof of the main theorem which is proved in [Y. Kamiyama, M. Tezuka and T. Toma, Homology of the configuration spaces of quasi-equilateral polygon linkages, Trans. Am. Math. Soc. 350, No. 12, 4869-4896 (1998)]. Let \(M_n\) be the configuration space of planar polygons \((u_1, \dots, u_n)\) whose vertices are \(u_i \in \mathbb{R}^2\) and whose edges have Euclidean length one. The main theorem (Theorem 1.1 in the paper) is to determine \(H_*(M_n, Z)\) which is torsion free and to calculate the Poincaré polynomial of \(M_n\) for \(n= 2m+1\) odd and \(n=2m\) even cases. The method of the proof is (1) to identify \(M_n\) as a complex algebraic variety instead of a real algebraic variety by \(z_i = u_{i+1} - u_i\), and the complement of \(M_n\) in \((S^1)^n\) is a manifold \(X_n\); (2) using a Morse function from the definition of \(M_n\), \(f: X_n \to \mathbb{R}\) to compute the critical points and their Morse indexes, one has the homotopy type of \(X_n\) and the Euler characteristic of \(M_n\) (basically the application of Lefschetz theorem on hyperplane sections for \(M_n\)); (3) \(M_n\) for \(n=2m\) even is a space with one singular point where the local neighborhood of the singular point can be described as a cone of \(S^{m-2} \times S^{m-2}\), by Poincaré-Lefschetz duality, one has that \(M_{2m}\) is a \((m-2)\)-regular \((2m-3)\)-space. The calculation splits in \(m-2\) for the homology groups.

MSC:

57M99 General low-dimensional topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
14P05 Real algebraic sets
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