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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some computer-assisted topological models of Hilbert fundamental domains

Author: Harvey Cohn
Journal: Math. Comp. 23 (1969), 475-487
MSC: Primary 10.21; Secondary 68.00
MathSciNet review: 0246820
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Abstract: The Hubert modular group $ {\text{H}}$ for the integral domain $ {\text{O}}({k^{1/2}})$ has a fourdimensional fundamental domain $ {\text{R}}$ which should be represented geometrically (like the classic modular group). Computer assistance (by the Argonne CDC 3600) was used for outlining cross sections of the three-dimensional "floor" of $ {\text{R}}$, which is a mosaic of an intractably large number of boundary pieces identified under $ {\text{H}}$. The cross sections shown here might well contain enough information when $ k = 2,3,5,6$ to form some "incidence matrices" and see $ {\text{R}}$ (at least) combinatorially. For special symmetrized subgroups of $ {\text{H}}$, it is plausible to see homologously independent $ 2$-spheres in (the corresponding) $ {\text{R}}$. The program is a continuation of one outlined in two earlier issues of this journal v. 19, 1965, pp. 594-605, MR 33 #4016, and v. 21, 1967, pp. 76-86, MR 36 #5081.

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Article copyright: © Copyright 1969 American Mathematical Society