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

 

Constructing product fibrations by means of
a generalization of a theorem of Ganea


Author: Paul Selick
Journal: Trans. Amer. Math. Soc. 348 (1996), 3573-3589
MSC (1991): Primary 55P99, 55P10
MathSciNet review: 1329539
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Abstract: A theorem of Ganea shows that for the principal homotopy fibration $\Omega B\to F\to E$ induced from a fibration $F\to E\to B$, there is a product decomposition $\Omega (E/F)\approx \Omega B\times \Omega (F*\Omega B)$. We will determine the conditions for a fibration $X\to Y\to Z$ to yield a product decomposition $\Omega (Z/Y)\approx X\times \Omega (X*Y)$ and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime $p$, homotopy decompositions for $\Omega {J_{k}\left (S^{2n}\right )}$ for some $k$ which include the cases $k=p^{t}$. The factors of $\Omega {J_{p^{t}}\left (S^{2n}\right )}$ consist of the homotopy fibre of the attaching map $S^{2np^{t}-1}\to {J_{p^{t}-1}\left (S^{2n}\right )}$ for ${J_{p^{t}}\left (S^{2n}\right )}$ and combinations of spaces occurring in the Snaith stable decomposition of $\Omega ^{2} S^{2n+1}$.


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Additional Information

Paul Selick
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 1A1
Email: selick@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01517-6
PII: S 0002-9947(96)01517-6
Received by editor(s): August 9, 1994
Additional Notes: Research partially supported by a grant from NSERC
Article copyright: © Copyright 1996 American Mathematical Society