Some computer-assisted topological models of Hilbert fundamental domains

Cohn, Harvey

doi:10.1090/S0025-5718-1969-0246820-9

Some computer-assisted topological models of Hilbert fundamental domains
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Math. Comp. 23 (1969), 475-487 Request permission

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.

References

Harvey Cohn, A numerical survey of the floors of various Hilbert fundamental domains, Math. Comp. 19 (1965), 594–605. MR 195818, DOI 10.1090/S0025-5718-1965-0195818-4
Harvey Cohn, A numerical study of topological features of certain Hilbert fundamental domains, Math. Comp. 21 (1967), 76–86. MR 222029, DOI 10.1090/S0025-5718-1967-0222029-8
Harvey Cohn, Sphere fibration induced by uniformization of modular group, J. London Math. Soc. 43 (1968), 10–20. MR 228435, DOI 10.1112/jlms/s1-43.1.10