On the topological classification of the floors of certain Hilbert fundamental domains

Abstract:

Associated to the field $Q({k^{1/2}})$ ($k$ a positive square free integer greater than one), there is a group of transformations of the product of two upper half planes which is analogous to the Hilbert modular group. This group has been shown to have a fundamental domain bounded by a finite number of hypersurfaces. Of particular interest is a subspace of the domain known as the “floor.” This floor is a quotient space of a fiber bundle over the circle which is determined by the field $Q({k^{1/2}})$. The principal result of this paper is that, conversely, the topological type (indeed the homotopy type) of this fiber bundle determines the field $Q({k^{1/2}})$ which gives rise to it. This is accomplished by computing the homology groups of the fiber space and showing that the integer $k$ can be determined from these groups.