The homology of Kummer manifolds

The theory of Kummer surfaces is a classical topic in algebraic geometry.1A generalization to higher dimensions has been given by W. Wirtinger.2The 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.4From 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 Í 1 ii j Es, jih coordinate of p, = < (-1 if j E s.The points p, are the points equal to their inverses under the group structure on T. Let G denote the group of transformations on T consisting of the identity and the map g: T^>T defined by g(x)=the inverse of x with respect to the group operation on T. For each 5 let B, denote an open 2«-dimensional ball about p" with boundary S,