260 G.M.L. Powell
Thus H
∗
A
m +1
∼ = H
∗
A
m
⊗ F
p
[y
m
]y
α
, so that the induction hypothesis on the degrees of the cohomology is satisfied.
This argument calculates the cohomology algebra H
∗
ΛW ; F
p
. This is: H
∗
Z; F
p
∼ = Λx
1
⊗
K
O
i =1
F
p
[y
i
]y
α
i
i
. To complete the proof of the proposition and the second statement of Theorem 1, it
remains to show the statement concerning p-formality. This follows from the form of the model constructed over F
p
, which is the tensor product of an exterior algebra by factors of the form Λx, y with dy = 0 and dx = y
α
. It may be seen that these factors correspond to p-formal spaces, using the techniques of [7], so that the tensor
product does as well.
3.1 The model over
Z
p
Using the above, it is possible to give the model ΛW, d with coefficients in Z
p
. Let B
m
be the sub-CGDA over Z
p
defined by B
m
= Λx
1
, y
1
, . . . , x
m
, y
m
, d, where the elements x
i
, y
i
are basis elements as in Proposition 3.1. There is no ambiguity here, since W has at most one element in each degree. Thus B
m +1
= B
m
⊗Λx
m +1
, y
m +1
, d, with B
m
as a sub-CGDA. Observe that W is in degrees ≥ 3 since W is connected and the lowest degree element of W must be in odd degree.
This shows that the differential of y
m +1
cannot involve x
m +1
; that is dx
m +1
= p
r
m+1
y
m +1
+ B
m
dy
m +1
= B
m
, where B
m
indicates decomposable elements of B
m
. Proposition 3.2 For all 1 ≤ m ≤ K the algebra H
∗
B
m
; Z
p
is concentrated in even degrees k|y
1
| for k ≥ 0. As an algebra it is generated by elements: {y
1
, . . . , y
m
} subject to the relations
p
r
1
y
1
= 0 p
r
j+1
y
j +1
+ y
α
j
j
= 0 for some integers r
j
≥ 1; there is a choice of indecomposables of B
m
so that the differential is:
dx
1
= p
r
1
y
1
dx
j +1
= p
r
j+1
y
j +1
+ y
α
j
j
for j K. Proof: The proof is by induction on m.
B
1
= Λx
1
, y
1
, dx
1
= p
r
1
y
1
, where the differential is forced to act as given, for degree reasons.
Suppose that the result is true for m ≤ M and consider B
M +1
= B
M
⊗Λy
M +1
⊗ Λx
M +1
, d. Now dy
m
is a cocycle of odd degree in B
M
, so it is the coboundary of a decomposable element in B
M
, by the hypothesis on the cohomology of B
M
over Z
p
. By changing the space of indecomposables if necessary, one may suppose that dy
M +1
= 0.
Elliptic spaces with the rational homotopy type of spheres 261
Knowledge of the structure of ΛZ, d ⊗ F
p
shows that |x
M +1
| + 1 = α
M
|y
M
| for some α
M
≥ 2. Now, by the inductive hypothesis on the structure of the cohomology algebra, the cohomology H
∗
B
M
is generated in degree |x
M +1
| + 1 by the class represented by the cocycle y
α
M
M
, so that H
∗
B
M
⊗ Λy
M +1
is generated as a Z
p
- module by y
α
M
M
and y
M +1
in that degree. Thus, again by changing the choice of indecomposable if necessary and absorbing
any unit multiples in Z
p
into the choice of generators, one may suppose that dx
M +1
= p
r
M +1
y
M +1
+ y
M α
M
. for some r
M +1
≥ 1. This proves the inductive step of the argument, since the homology of H
∗
B
M +1
may be calculated and it satisfies the statement of the Proposition.
To complete the determination of the model ΛW, d, one may show via the same arguments that there is a choice of indecomposable representing w
with differential dw
= y
α
K
K
. Thus, the minimal model M = ΛW, d has a choice of space of inde- composables over Z
p
for which W has a free basis: {x
1
, . . . , x
K
, }, {y
1
, . . . , y
K
}, w with respect to which the differential is:
dx
1
= p
r
1
y
1
dx
j +1
= p
r
j+1
y
j +1
+ y
α
j
j
for j K dw
= y
α
K
K
where r
j
≥ 1 and α
j
≥ 2 for all j.
3.2 Formality of the space
