THE 3-TORUS IS KERVAIRE 3
The problem can be recast geometrically
([H1]).
Let
K
be a CW-complex with
7ri
(K)
=
G
and form a new complex as follows. Corresponding to each generator
xi
of
F
attach a 1-cell to (the base point of)
K,
to give a complex
K'
with
7ri (
K')
=
G *F.
Now attach a 2-cell
aj
to
K'
with attaching map in the class of
Wj,
for j
=
1, · · · , m. This gives a complex
L
with
7ri(L)
=
(G*F)/N.
Moreover
{WI,··· , wm}
is Kervaire if and only if the canonical map from
7ri (
K)
to
7ri (
L)
is an injection.
The chain map
C2 (L, K)
-----+
CI (L, K),
is given, with respect to the obvious bases,
by
(di,j),
so
{wi,·· · ,wm}
is non-singular if and only if
H2 (L,K)
=
0.
Conversely given a relatively 2-dimensional pair of 2-complexes
(K, L)
then
(L/ K)(l)
generates a free group
F
and the 2-cells of
L\K
determine a set of words
{WI,· .. , wm}
in
7ri
(K)
*
F.
In this case we say that the system of equations
WI
=
1, · · · ,
Wm
=
1 is
realized
by
(K, L).
If
X
is a CW-complex and (K,L) is a
relatively 2-dimensional pair of 2-complexes with
L/ K
=
X
then we say that the
system of equations realized by (
K, L)
is
modelled
on
X.
A 2-complex
X
is said to
be
K ervaire
if Property A holds for every system of equations modelled on
X.
THE 3-TORUS
The 3-dimensional torus
SI
X
si
X
si
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-Iy-I,
yzy-Iz-I,zxz-Ix-I
.
We will show that this 2-complex K is Kervaire. This was proved by Gersten
[G2] in the case where
G
is torsion-free.
Fix a group G, and let
a,
b, c, d,
e,
J,
g,
h, i, j, k, l be elements of G. We must find
a group H, containing Gas a subgroup, and elements
x,
y,
z
E H, such that
We distinguish several cases, according to the values in
G
of the eight elements
a
=
eai,
(3
=
lbe-I,
'Y
=
f-Ie[-
I,
8
=
i-Idj, a'
=
h-Iaj-I, (3'
=
k-Ibh, "(1
=
gck,
and
8'
=
jdg-I,
which we will call the
corner labels.
The
star graph
of the system
(see [BP,EH,G2]) is the 1-skeleton of the octahedron (the dual of the cube by which
the 3-cell of the torus is attached), and these eight elements are the labels of the
eight triangular faces (or, equivalently, the corner labels of the eight corners of the
cube). We associate them with the corners of the cube according to the scheme
depicted in Figure 1. In general, some or all of these elements may be trivial in
G.
The different cases we consider reflect the numbers of different nontrivial corner
labels.
Theorem 1.
Suppose there is
a
pair of opposite
faces
of the cube of Figure 1, such
that
two
or more
of
the corner labels
on
each of these
faces are
nontrivial elements
of G. Then the system of equations has
a
solution in
an
overgroup of G.
Previous Page Next Page