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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of the non-triviality theorem of Serre


Author: Stephan Klaus
Journal: Proc. Amer. Math. Soc. 130 (2002), 1249-1256
MSC (1991): Primary 19D06, 20J05, 20J06, 55P20, 55P40, 55P42, 55P60, 55Q52, 55R35, 55S45, 55T10
Published electronically: December 20, 2001
MathSciNet review: 1879944
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Abstract: We generalize the classical theorem of Serre on the non-triviality of infinitely many homotopy groups of $1$-connected finite CW-complexes to CW-complexes where the cohomology groups either grow too fast or do not grow faster than a certain rate given by connectivity. For example, this result can be applied to iterated suspensions of finite Postnikov systems and certain spaces with finitely generated cohomology ring. In particular, we obtain an independent, short proof of a theorem of R. Levi on the non-triviality of $k$-invariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension $n$, $2n$, $3n, \ldots$ for a fixed number $n$. We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups.


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Additional Information

Stephan Klaus
Affiliation: Mathematisches Forschungsinstitut Oberwolfach, Lorenzenhof, 77709 Oberwolfach-Walke, Germany
Email: klaus@mfo.de

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06441-3
PII: S 0002-9939(01)06441-3
Keywords: Homotopy group, Postnikov system, $k$-invariant, Poincar\'e series, growth rate, suspension, Hilbert-Serre theorem, group cohomology, perfect group, $p$-completion, Quillen plus construction, stable homotopy group
Received by editor(s): January 6, 2000
Received by editor(s) in revised form: July 26, 2000
Published electronically: December 20, 2001
Communicated by: Ralph Cohen
Article copyright: © Copyright 2001 American Mathematical Society