The homology of Kummer manifolds
The homology of Kummer manifolds
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Kummer流形的同调性
DOI：
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发表时间：
1956
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通讯作者：
E. Spanier
中科院分区：
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作者：
E. Spanier
库默曲面理论是代数几何中的经典课题。Wirtinger。2品种介绍了Wirtinger可能被称为库默品种和代数维数n与22 n普通的双重点。如果这些双点以标准的方式通过“交换”去奇异化，则得到非奇异簇，其基础流形将被称为复维数n（拓扑维数2n）的库默流形。库默簇的挠问题最近已由A.安德烈奥蒂;[3]然而，由于这位作者没有对所研究的变种去奇异化，他的结果没有回答它们是否承认非分歧覆盖的基本问题。这个问题将在这里得到解决，通过显示，上述desingularized模型是简单的连接;此外，它的同调群将确定为所有维。4从纯拓扑的观点来看，所有库默流形的一个给定的层面是同胚的，可以定义如下。设T表示2 n维环面（n ^ 2），该环面被视为绝对值为1的复数与其自身的2 w倍乘积。对于整数1，...，...，2n的子集s，令Pi表示T的由下式定义的点：
The theory of Kummer surfaces is a classical topic in algebraic geometry.1 A generalization to higher dimensions has been given by W. Wirtinger.2 The varieties introduced by Wirtinger may be called Kummer varieties and have algebraic dimension n with 22n ordinary double points. If these double points are desingularized in a standard manner by means of "dilatations," one obtains nonsingular varieties whose underlying manifolds will be called Kummer manifolds of complex dimension n (topological dimension 2n). The question of the torsion of Kummer varieties has been discussed recently by A. Andreotti;3 since this author, however, did not desingularize the varieties under consideration, his results give no answer to the basic question whether they admit nonramified coverings or not. This question will be settled here by showing that the desingularized model mentioned above is simply connected ; moreover, its homology groups will be determined for all dimensions.4 From a purely topological point of view all Kummer manifolds of a given dimension are homeomorphic to one another and may be defined as follows. Let T denote the 2«-dimensional torus (n ^ 2) regarded as the 2w-fold product of the complex numbers of absolute value one with itself. For a subset s of the integers 1, ■ ■ ■ , 2n let p, denote that point of T defined by