Contemporary Mathematics
Volume 164, 1994
THE 3-TORUS IS KERVAIRE
ANDREW J. DUNCAN AND JAMES HOWIE
ABSTRACT.
The 3-dimensional torus 8
1
x 8
1
x 8
1
has a canonical cell-structure, with
one cell in dimensions 0 and 3, and three cells in dimensions 1 and 2. Its 2-skeleton
is thus the geometric realization of the group presentation
x, Y,z,
1
xyx-
1 Y- 1
,
yzy-
1
z-
1
,zxz-
1
x-
1
.
We show that this 2-complex K is Kervaire , in other words that any system of
equations over a group G, modelled on K, has a solution in some larger group.
INTRODUCTION
In this note we consider a complex associated to the 3-Torus 8
1
x 8
1
x 8
1
and
show that any system of equations over a group G, modelled on this complex has
a solution in some larger group. This means that the conclusion of the Kervaire-
Laudenbach conjecture holds for such a system of equations. In the remainder of
this section we outline the Kervaire-Laudenbach conjecture and in the following
section prove our result.
An early and general statement of the "Kervaire-Laudenbach Conjecture" is the
Adjunction Problem of B.H. Neumann [N]. This asks when the following property
holds.
Property A. Let G be a group and F the free group on generators x1
, · · · , Xn.
If
w1, · · · ,
Wm
are elements of G
*
F then the canonical map
G___.G*F
N
is injective, where N denotes the normal closure of
w
1
, · · · , Wm
in G *F.
Given words w1, · · · ,
Wm
in the free product
G
*
F
we may consider the expres-
sions w1 = 1, · · · ,
Wm
= 1 as a system of equations in unknowns
Xi
with coefficients
from G. This system of equations is said to have a solution over G if there exists
a group Hand a map¢:
G
*
F _____. H
such that ¢(wi) = 1, fori= 1, · · · , m, and
¢Ia is an injection. Since every such map¢ factors through (G
*
F)/N, Property
A is equivalent to the statement that every system of equations has a solution over
G.
1980 Mathematics Subject Classification (1985 Revision). Primary 20F05; Secondary 20F32.
Research supported by SERC grant GR/E 88998 .
This paper is in final form and no version of it will be submitted for publication elsewhere.
1
©
1994 American Mathematical Society
0271·4132/94 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/164/01579
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