The Homology of Singular Polygon Spaces

Kamiyama, Yasuhiko

Canadian journal of mathematics, Tome 50 (1998) no. 3, pp. 581-594

Résumé

Let ${{M}_{n}}$ be the variety of spatial polygons $P\,=\,({{a}_{1}},\,{{a}_{2}},...,{{a}_{n}})$ whose sides are vectors ${{a}_{i}}\,\in \,{{\mathbf{R}}^{3}}$ of length $\left| {{a}_{i}} \right|\,=\,1\,(1\,\le \,i\,\le \,n)$ , up to motion in ${{\mathbf{R}}^{3}}$ . It is known that for odd $n$ , ${{M}_{n}}$ is a smooth manifold, while for even $n$ , ${{M}_{n}}$ has cone-like singular points. For odd $n$ , the rational homology of ${{M}_{n}}$ was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of ${{M}_{n}}$ for even $n$ . For even $n$ , let ${{\tilde{M}}_{n}}$ be the manifold obtained from ${{M}_{n}}$ by the resolution of the singularities. Then we also determine the integral homology of ${{\tilde{M}}_{n}}$ .

Détail

DOI : 10.4153/CJM-1998-032-6

Mots-clés : 14D20, 57N65

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