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.

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.