COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

380 Harvey Cohn
always dictated by the symmetries of the faces in Fig. 2. As S goes from
0 to 1 the cross-section becomes more and more oblique, attaching itself
and detaching itself from images of the spindle (of norm 2) at critical
values S=O.31 (approx.) and S=O.82 (approx.). For the piece of norm 1,
we can show every transformation joins ,&, ie. for H in P
zo = H(-l/z). (4.9)
We use shading in Fig. 4 to show the portion which belong to r;.
Thus in terms of H in r;, we have the following:
“shaded portion” z. = H(- l/z),
“blank portion” z. = H(- l/z+ 1).
The shading is not of topological interest as much as it shows that part of
the piece of norm 1 must match equivalent points in the neighboring replica
of @- (formed by zf 1 = zo), if we were to match this piece in the
unit square by (4.9). Actually, it is more meaningful to match it with
itself.
The piece of norm 1, as reassembled for Fig. 4, is matched with itself under
zo = - l/z (or zi = - 1 /z) without use of the equivalence operations of (4.9).
To see this consider the two-dimensional boundaries of the reassembled
piece of norm 1. They consist of the simultaneous equations
IIZII = 1, II Yz+a II = 1 (4.10)
if ‘yz+ 6 is the denominator of a neighboring region. Under z. = - 1 /z,
the boundary is mapped into another boundary given by
IIZ II = 19 11 6z-y 11 = 1. (4.11)
The assertion now follows from the fact that the relation z. = - l/z converts
the piece into another piece of the same floor of norm 1, while the
boundary was determined in an invariant fashion.
Any boundary segment (4.10) determines the height of the floor uniquely.
These correspond to the points A and C in Fig. 1 which are determined
by fixed points rather than by any analysis of the arc ABC. Note
that the pair of points A, C constitutes a O-sphere just as the boundary
of the piece of norm 1 is a 2-sphere.
The critical values of S are quite interesting by themselves, namely
and
S1 = 0.3101 . . . = 4-6: /5
c 1
S2 = 0.8165 . . . = 6’,3
Actually the points of attachment and detachment are points at which S’
takes the values believed to be the minimum for the whole floor, namely [l]
S’ = +(-3+2.6’) = 0.4747 . . . .
Algebraic topology on bicomplex manifolds
5. Approximate topological configuration. By using the previous information
we can give an approximate description of the topological configuration.
First consider @. Here @ has a manifold point at 00 from which the threedimensional
base in Fig. 2 appears like a 3-sphere. It is divided into two
pieces of norms 1 and 2 which are 3-cells, each folded into itself by transformations
(4.8) and (4.9). (Recall that in the simple case, Fig. 1, the floor was
one piece folded onto itself by z. = - 1 /z.)
Next consider %. If we refer again to Fig. 1, we see that there are three
replicas of the floor AC, DC, EC which transform into one another like the
representatives of G/G2 in (2.7). The fact that the reassembled piece of norm
1 is transformed into itself under z. = - l/z, etc., enables us to reproduce
three replicas of that piece in an analogue of Fig. 1. If we momentarily
restrict ourselves to this piece (ignoring that of norm 2), we have a simple
situation where the (spherical) boundaries of each of the three replicas meet
in a 2-sphere analogous to the O-sphere A, C of Fig. 1. The analogy, however,
is not kept because the piece of norm 2 does not map into itself under
zo = - l/z, etc., hence it is not representable as three replicas lying in the
replicas of the floor. The boundary of this piece of norm 2 is nestled, however,
in between the various boundaries of the pieces of norm 1, and the
three-dimensional piece of norm 2 bulges out of the spherical boundary into
parts of the three-dimensional floor. The situation is therefore somewhat
more complicated than that of lower dimensional space (as it must be since
the final configuration cannot be a 4+phere!).
We are confronted with the need to study the self-mapping of the floor
more carefully in order to make deductions concerning the topology of the
fundamental domains @and @e. There are very few cases which are analogous
[5], possibly the other two involve Q(31j2), where the problem can be
considered as an analogous pasting of 3 (or 4) 3-spheres. In any case, the
numerical data are capable of further analysis for geometric or topological
features than attempted here.
REFERENCES
1. H. COHN: A numerical study of the floors of various Hilbert fundamental domains.
Math. of Compuration 19 (1965), 594-605.
2. H. COHN: A numerical study of topological features of certain Hilbert domains. Math.
of Computation 21 (1967), 76-86.
3. H. COHN : Conformal Mapping on Riemann Surfaces, McGraw Hill, 1967.
4. K. B. GIJNDLACH: Some new results in the theory of Hilbert’s modular group,
pp. 165-180, Contributions to Function Theory (Tata Institute, 1960).
5. K. B. GUNDLACH: Die Fixpunkte einiger Hilbertscher Modulgruppen. Math. Annalen
157 (1965), 369-390.
6. W. M. W OODRUFF: The singular points of the fundamental domain for the groups of
of Bianchi, Ph.D. Dissertation, University of Arizona, 1967.
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