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

• Published in: Mathematische Zeitschrift Vol. 273; no. 3-4; pp. 735 - 751
• Main Authors: Beben, Piotr, Wu, Jie
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2013
• Subjects: Mathematics; Mathematics and Statistics; Finite; Modular representations; Loop space decompositions; complexes
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-012-1027-7

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 of X , we use a decomposition of ΩΣ X due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ΩΣ i X is a homotopy retract of ΩΣ X . This implies the stable homotopy groups of Σ X are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for Σ X . Under additional assumptions on , we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ΩΣ X that has infinitely many finite H -spaces as factors.