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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.

Reviewer: Weiping Li (Stillwater)