22
MARTIN ARKOWITZ AND GREGORY LUPTON
for each basis element Wj of V1.
We next show that this restricted obstruction is in fact the obstruction to
f
and f' being homotopic. This situation is comparable to that of Proposition 4.7,
and indeed the proofs are similar.
7.3 Proposition
Let
J,
f': M----+ M be maps with[!], [!']
E
E#(M) and !Iva
=
f'
I
v0
=
tl
v0

Then f
c:::
f' if and only if the restricted obstruction 0
1
(!,
f')
=
0.
Proof;:::::
If
cit(!,!')
=
0, then
{,.,
=
./,.,
for each
j
and for each appropriate
E
and
'f).
We define a homotopy
H
starting at
f
and ending at a map
f"
homotopic
to
f'. Set Hlv
=
f and Hlv
=
0. On
V,
set
Hlv,
=
0, H(iii)
=
0 and
H(v)
= '"'
l(,'j - ,]
)uEvT/' · · · vT/1
· ·
vT/q
J ~ 2 I""E,T/
I""E,rf
1 J
q '
E,T/
T/j=1
where
vT
indicates that this term is omitted. From Lemma 3.2(ii), we have that
a(wj)
=
Wj
+
Wj
+
2vjVj
+
Zj with Zj
E
(V0
).
By construction, H(V)
=
0.
Furthermore it is easily seen that
H
ends at a map
!"
=
H
a
that satisfies
f"
I
v0 =
tlva and
for each
j.
Therefore 01 ((,
f') =
0 :
V1
----+
H* (M), and so by Proposition 3.3(ii),
!"
c:::
f'.
=? :
On the other hand, suppose
H
is a homotopy that starts at
f
and ends at f'.
By Lemma 4.3, Hlva
=
0.
From Lemma 3.2(ii), we have,
f'(wj)
=
Ha(wj)
=
f(wj)
+
dH(wj)
+
2vjH(vj),
for each
j.
Since f(wj)- f'(wj)
=
-dH(wj), modulo terms in the ideal (vj), it
follows that .~.T/
=
)..~~T/ and hence that 01
(!,
f')
=
0.
D
Proposition 7.3 allows us to give necessary and sufficient conditions for the
product of spheres X to have E*#(X) infinite. Recall that
m
=
(m1, ... ,mp) and
n
= (
n
1
, ... ,
nq) and consider the following numerical condition:
7.4 Condition
There is a binary p-tuple
E
and a binary q-tuple
'f)
with
'f)j
=
0
for
some
j
such that
lEI
+
lrJI
2:
2 and
E ·
m
+
rJ · n
=
2nj -
1.
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