THE 3-TORUS IS KERVAIRE
7
Theorem 4.
The standard 2-skeleton K of the 3-torus is Kervaire.
Proof.
By the Corollary to Theorem 1, we may assume that we have at most four
nontrivial corner labels. Moreover, if there are precisely four, then by Theorem 1
the result still follows unless there is no pair of parallel faces, each containing two of
the nontrivial corner labels. Given that precisely four corner labels are non-trivial
assume, for
the sake of argument that o/ is non-trivial. Then if
!3' =
8'
=
o:
=
1,
either
j3
=
1, in which case the top and bottom faces each have two non-trivial
labels, or
j3
-1-
1 in which case the front and back faces each have two non-trivial
labels. Hence we may assume the non-trivial corner labels occur in pairs connected
by edges.
It
then follows from Theorem 1, again, that we may assume that the
four nontrivial labels are distributed at one vertex and its three neighbours, say
o:, o:',
!3',
8'. But then a change of variable move as described above changes our
system to one where the four nontrivial labels are distributed as
o:', j3', 8', 8,
and
the result again follows from Theorem 1.
Suppose next that precisely three corner labels are nontrivial.
If
they all belong
to a common face of the cube, we may apply Theorem 3, so suppose that this does
not happen.
If
two of the nontrivial labels are at adjacent corners, then we may
assume without loss of generality that the three nontrivial labels are
o:, f],
8'. Then
the change of variable move gives us a system to which we may apply Theorem 3.
Finally, if no two of the three labels lie at adjacent corners, we may assume they
are
o:,
!3',
8'. The change of variable move reduces us to the previous case, where
two nontrivial labels are adjacent.
Finally, suppose at most two labels are nontrivial. Then Theorem 3 applies
except in the case where these labels are at opposite corners, say
o:, "(
1

Apply the
change of variable move once again to produce a system with two nontrivial labels
on a common face, and then apply Theorem 3.
Remarks. In the case of no nontrivial corner labels, the solution to the system is
trivial. As pointed out by Gersten [G2], if one of the corner labels is nontrivial,
then so is at least one more.
If
precisely two labels are nontrivial, then standard
arguments reduce us to the case where G is cyclic. In this case, an alternative
argument can be given: construct a central extension of
G
by
Z
3
in which the
equations hold, namely that corresponding to the 2-cocycle (abed, efgh, ijkl)
E
Q3
~
H2(z3, G).
REFERENCES
[BP] W. Bogley and S.
J.
Pride, Aspherical relative presentations, Proc. Edinburgh Math. Soc.
35 (1992), 1-39.
[B] A. Dold and B. Eckmann (eds), Proceedings of the Second International Conference on the
Theory of Groups, Australian National University, Canberra, 1973, Lecture Notes in Mathe-
matics, 372, Springer, Berlin; New York, 1974.
[E1] M. Edjvet, The solution of certain sets of equations over groups, Groups-St. Andrews, 1989
(C.M. Campbell and E.F. Robinson, eds.), London Math. Soc. Lecture Notes Series, vol. 159,
Cambridge Univ. Press, 1991, pp. 105-123.
[E2] M.Edjvet, Equations over Groups and a Theorem of Higman, Neumann and Neumann, Proc.
London Math. Soc. (3) 62 (1991), 563-589.
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