It is a classical result of Postnikov [15]
that homotopy types X with at most two non-trivial homotopy groups
πmX = A and
πnX = B, 2 [les ] m < n,
are classified by the k-invariant
formula here
Here the cohomology group of the Eilenberg–MacLane space
K(A, m) was computed
by Eilenberg–MacLane [11] and Cartan [5].
Let p be a prime and let ℤp ⊂ ℚ be the
smallest subring of ℚ containing 1/q for all primes
q with q ≠ p. We consider finitely
generated ℤp-modules A and B
and the stable range n < 2m − 1. Hence X is a
p-local space with at most two non-trivial homotopy groups in a stable range. Then
the homotopy type of X admits a product decomposition
formula here
where all Xi with 1 [les ] i [les ] j
are indecomposable and this decomposition is unique up
to permutation. We classify in this paper the indecomposable factors in (2) by the
following result.