On the cohomology of torus manifolds
On the cohomology of torus manifolds
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DOI：
10.18910/12823
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发表时间：
2003-06
影响因子：
0.4
通讯作者：
M. Masuda;T. Panov
中科院分区：
文献类型：
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作者：
M. Masuda;T. Panov
环面流形是由具有非空不动点集和一些附加符号数据的半维环面作用的偶维流形。它可以被认为是对代数几何中的环流形的一个深远的推广。环面流形的轨道空间具有丰富的组合结构，如在局部标准作用下，它是一个带角的流形。本文研究了环面流形的上同调性质与其轨道商的组合之间的关系。证明了环面流形的上同调环是由二维类生成的，当且仅当商是同调多面体。在这种情况下，我们从环形几何中获得熟悉的图像：等变上同调是神经简单复核的面环，普通上同调是通过分解某些线性形式得到的。在更一般的情况下，我们证明了环面流形的奇次上同调当且仅当轨道空间是面无环时消失。虽然在这种情况下不再产生二阶上同调，但等变上同调仍然与适当的简单偏序集的面环同构。
A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orie ntation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geo metry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.