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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homology and some homotopy decompositions
for the James filtration on spheres


Author: Paul Selick
Journal: Trans. Amer. Math. Soc. 348 (1996), 3549-3572
MSC (1991): Primary 55P99, 55P10
MathSciNet review: 1348157
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Abstract: The filtrations on the James construction on spheres, $ J_{k}\left (S^{2n}\right )$, have played a major role in the study of the double suspension $S^{2n-1}\to \Omega ^2 S^{2n+1}$ and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate $ H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb {Z}}/p{\mathbb {Z}}\right )$ for odd primes $p$. When $k$ has the form $p^t-1$, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher $p$-torsion. The calculations are applied to construct a $p$-local product decomposition of $ \Omega J_{k}\left (S^{2n}\right )$ for $k<p^2-p$ which demonstrates a mod $p$ homotopy exponent in these cases.


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Additional Information

Paul Selick
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1
Email: selick@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01593-0
PII: S 0002-9947(96)01593-0
Received by editor(s): July 21, 1994
Additional Notes: Research partially supported by a grant from NSERC
Article copyright: © Copyright 1996 American Mathematical Society