The cohomology ring of polygon spaces
The cohomology ring of polygon spaces
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多边形空间的上同调环
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
A. Knutson
中科院分区：
文献类型：
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作者：
J. Hausmann;A. Knutson
本文计算了[豪斯曼，Klyachko，Kapovich-Millson]中引入的“多边形空间”的整数上同调环.这是通过将它们嵌入到某些环面簇中来完成的;上同调的限制映射是满射的，我们使用Gr“obner基理论的思想来计算它的核。由于我们不反转素数2，我们可以张量为Z/2;将所有度减半，我们证明这产生了平面多边形空间的Z/2上同调环。在等边的情况下，有一个作用的对称群置换的边缘，我们表明，诱导作用的整数上同调不是标准的，尽管它是这样的有理上同调[K1]。最后，我们的庞加莱多项式的公式是更有效的计算比那些已知的[K1]。
We compute the integer cohomology rings of the ``polygon spaces' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Gr"obner bases. Since we do not invert the prime 2, we can tensor with Z/2; halving all degrees we show this produces the Z/2 cohomology rings of planar polygon spaces. In the equilateral case, where there is an action of the symmetric group permuting the edges, we show that the induced action on the integer cohomology is _not_ the standard one, despite it being so on the rational cohomology [Kl]. Finally, our formulae for the Poincar'e polynomials are more computationally effective than those known [Kl].