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Proceedings of the American Mathematical Society

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On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

Author: Alexander N. Dranishnikov
Journal: Proc. Amer. Math. Soc. 137 (2009), 1489-1497
MSC (2000): Primary 55M30
Published electronically: November 25, 2008
MathSciNet review: 2465675
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Abstract: The following inequality

$\displaystyle \mathrm{cat}_{\mathrm{LS}} X\le\mathrm{cat}_{\mathrm{LS}} Y+\bigg\lceil\frac{hd(X)-r}{r+1}\bigg\rceil$

holds for every locally trivial fibration $ f:X\to Y$ between $ ANE$ spaces which admits a section and has the $ r$-connected fiber, where $ hd(X)$ is the homotopical dimension of $ X$. We apply this inequality to prove that

$\displaystyle \mathrm{cat}_{\mathrm{LS}} X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$

for every complex $ X$ with $ cd(\pi_1(X))\le 2$, where $ cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $ X$.

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Additional Information

Alexander N. Dranishnikov
Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, Florida 32601-8105

Received by editor(s): September 25, 2007
Received by editor(s) in revised form: April 27, 2008
Published electronically: November 25, 2008
Additional Notes: The author was supported by NSF grant DMS-0604494
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.