A generalization of the non-triviality theorem of Serre
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- by Stephan Klaus
- Proc. Amer. Math. Soc. 130 (2002), 1249-1256
- DOI: https://doi.org/10.1090/S0002-9939-01-06441-3
- Published electronically: December 20, 2001
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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.References
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Bibliographic Information
- Stephan Klaus
- Affiliation: Mathematisches Forschungsinstitut Oberwolfach, Lorenzenhof, 77709 Oberwolfach-Walke, Germany
- Email: klaus@mfo.de
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1249-1256
- MSC (1991): Primary 19D06, 20J05, 20J06, 55P20, 55P40, 55P42, 55P60, 55Q52, 55R35, 55S45, 55T10
- DOI: https://doi.org/10.1090/S0002-9939-01-06441-3
- MathSciNet review: 1879944