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Modular representations and the homotopy of low rank p-local CW-complexes

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Abstract

Fix an odd prime p and let X be the p-localization of a finite suspended CW-complex. Given certain conditions on the reduced mod-p homology \({\widetilde H_*(X; \mathbb{Z}_p)}\) of X, we use a decomposition of ΩΣX due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ΩΣi X is a homotopy retract of ΩΣX. This implies the stable homotopy groups of ΣX are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for ΣX. Under additional assumptions on \({\widetilde H_*(X; \mathbb{Z}_p)}\), we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ΩΣX that has infinitely many finite H-spaces as factors.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Si Yuan Building, Beijing, 100190, People’s Republic of China

    Piotr Beben

  2. Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10, Lower Kent Ridge Road, Singapore, 119076, Singapore

    Jie Wu

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  1. Piotr Beben
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  2. Jie Wu
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Correspondence to Piotr Beben.

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Beben, P., Wu, J. Modular representations and the homotopy of low rank p-local CW-complexes. Math. Z. 273, 735–751 (2013). https://doi.org/10.1007/s00209-012-1027-7

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