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

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 cat1(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) dim(Bπ1(X)) + cat1(X), which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].

Lusternik–Schnirelmann category, skeleta, fundamental group, symplectic manifold
Mathematical Subject Classification 2000
Primary: 55M30
Secondary: 55P99
Received: 15 December 2009
Revised: 22 April 2010
Accepted: 24 April 2010
Published: 23 May 2010
Corrected: 11 August 2010
John Oprea
Department of Mathematics
Cleveland State University
Cleveland OH
Jeff Strom
Department of Mathematics
Western Michigan University
Kalamazoo MI