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