Volume 10, issue 2 (2010)

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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
Abstract

In this paper, we study the growth with respect to dimension of quite general homotopy invariants $\mathsc{Q}$ 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 $\mathsc{A}$–category. We use ${cat}^{1}\left(X\right)$ (which is $\mathsc{A}$–category with $\mathsc{A}$ 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 $cat\left(X\right)\le dim\left(B{\pi }_{1}\left(X\right)\right)+{cat}^{1}\left(X\right)$, 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
Primary: 55M30
Secondary: 55P99
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/