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.