Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

Oprea, John; Strom, Jeff

doi:10.2140/agt.2010.10.1165

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Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

John Oprea and Jeff Strom

Algebraic & Geometric Topology 10 (2010) 1165–1186

DOI: 10.2140/agt.2010.10.1165

Abstract

In this paper, we study the growth with respect to dimension of quite general homotopy invariants $Q$ applied to the CW skeleta of spaces. This leads to upper estimates analogous to the classical “dimension divided by connectivity” bound for Lusternik–Schnirelmann category. Our estimates apply, in particular, to the Clapp–Puppe theory of $A$ –category. We use ${cat}^{1} (X)$ (which is $A$ –category with $A$ the collection of $1$ –dimensional CW complexes), to reinterpret in homotopy-theoretical terms some recent work of Dranishnikov on the Lusternik–Schnirelmann category of spaces with fundamental groups of finite cohomological dimension. Our main result is the inequality $cat (X) \leq dim (B π_{1} (X)) + {cat}^{1} (X)$ , which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].

Keywords

Lusternik–Schnirelmann category, skeleta, fundamental group, symplectic manifold

Mathematical Subject Classification 2000

Primary: 55M30

Secondary: 55P99

References

Publication

Received: 15 December 2009

Revised: 22 April 2010

Accepted: 24 April 2010

Published: 23 May 2010

Corrected: 11 August 2010

Authors

John Oprea
Department of Mathematics Cleveland State University Cleveland OH 44115
http://academic.csuohio.edu/oprea_j/
Jeff Strom
Department of Mathematics Western Michigan University Kalamazoo MI 49008-5200
http://homepages.wmich.edu/~jstrom/

Volume 10, issue 2 (2010)