The Homology of Singular Polygon Spaces
The Homology of Singular Polygon Spaces
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奇异多边形空间的同调
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
Y. Kamiyama
中科院分区：
文献类型：
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作者：
Y. Kamiyama
Abstract 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 $left| {{a}_{i}}
ight|,=,1,(1,le ,i,le ,n)$ , 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 ${{ ilde{M}}_{n}}$ be the manifold obtained from ${{M}_{n}}$ by the resolution of the singularities. Then we also determine the integral homology of ${{ ilde{M}}_{n}}$ .
Abstract 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 $left| {{a}_{i}}
ight|,=,1,(1,le ,i,le ,n)$ , 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 ${{ ilde{M}}_{n}}$ be the manifold obtained from ${{M}_{n}}$ by the resolution of the singularities. Then we also determine the integral homology of ${{ ilde{M}}_{n}}$ .