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The Serre spectral sequence of a multiplicative fibration


Authors: Yves Félix, Stephen Halperin and Jean-Claude Thomas
Journal: Trans. Amer. Math. Soc. 353 (2001), 3803-3831
MSC (2000): Primary 57T25, 55R20, 57T05
DOI: https://doi.org/10.1090/S0002-9947-01-02801-X
Published electronically: April 24, 2001
MathSciNet review: 1837260
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Abstract:

In a fibration $\Omega F \overset{\Omega j}{\rightarrow} \Omega X \overset{\Omega \pi}{\rightarrow}\Omega B$ we show that finiteness conditions on $F$ force the homology Serre spectral sequence with $\mathbb{F} _p$-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, $H_*(\Omega X)$ is ``almost'' isomorphic to $H_*(\Omega B)\otimes H_*(\Omega F)$. One consequence is the conclusion that $X$ is elliptic if and only if $B$ and $F$ are.


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Additional Information

Yves Félix
Affiliation: Institut de Mathématiques, Université de Louvain-la-Neuve, B-1348 Louvain-la- Neuve, Belgium

Stephen Halperin
Affiliation: College of Computer, Mathematical and Physical Sciences, University of Maryland, College Park, Maryland 20742-3281

Jean-Claude Thomas
Affiliation: Faculté des Sciences, Université d’Angers, 49045 bd Lavoisier, Angers, France

DOI: https://doi.org/10.1090/S0002-9947-01-02801-X
Keywords: Multiplicative fibration, loop space, Hopf algebra, Serre spectral sequences, elliptic space
Received by editor(s): January 30, 1998
Published electronically: April 24, 2001
Additional Notes: Research of the second author was partially supported by an NSERC operating grant. Research of the first and third authors was partially supported by UMR-6093 au CNRS
This work was also partially supported by a NATO travel grant held by all three authors.
Article copyright: © Copyright 2001 American Mathematical Society

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