Remote Access Mathematics of Computation
Green Open Access

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
DOI: https://doi.org/10.1090/S0025-5718-1969-0246820-9
MathSciNet review: 0246820
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [6] H. Cohn, "A numerical survey of the floors of various Hilbert fundamental domains," Math. Comp., v. 19, 1965, pp. 594-605. MR 33 #4016. MR 0195818 (33:4016)
  • [12] H. Cohn, "A numerical study of topological features of certain Hilbert fundamental domains," Math Comp., v. 21, 1967, pp. 76-86. MR 36 #5081. MR 0222029 (36:5081)
  • [13] H. Cohn, "Sphere fibration induced by uniformization of modular group," J. London Math. Soc., v. 43, 1968, pp. 10-20. Erratum in [6] p. 603, Table 3. For $ k = 13$, $ \vert N(\gamma )\vert = 4({\text{not}}3)$ (not 3). Errata in [12] p. 81, Figure 2. Instead of $ S$, read $ {S_1}$. p. 82, Figure 3. Instead of $ S = - .80$, read $ {S_1} = - .40$, $ S = - .60$, $ {S_1} = - .20$, $ S = - .40$, $ {S_1} = 0.00$. pp. 84-85, Figure 4. Instead of $ S$, read $ {S_1}$. MR 0228435 (37:4015)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10.21, 68.00

Retrieve articles in all journals with MSC: 10.21, 68.00


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1969-0246820-9
Article copyright: © Copyright 1969 American Mathematical Society

American Mathematical Society