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Representation theory of homotopy types with at most two non-trivial homotopy groups localized at a prime

Published online by Cambridge University Press:  01 March 2000

HANS-JOACHIM BAUES  and
YURI DROZD
HANS-JOACHIM BAUES
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
YURI DROZD
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 128 , Issue 2 , March 2000 , pp. 283 - 300
Copyright
© The Cambridge Philosophical Society 2000

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