Algebraic & Geometric Topology

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

John Oprea and Jeff Strom

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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].

Article information

Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1165-1186.

Received: 15 December 2009
Revised: 22 April 2010
Accepted: 24 April 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space
Secondary: 55P99: None of the above, but in this section

Lusternik–Schnirelmann category skeleta fundamental group symplectic manifold


Oprea, John; Strom, Jeff. Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov. Algebr. Geom. Topol. 10 (2010), no. 2, 1165--1186. doi:10.2140/agt.2010.10.1165.

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