## Algebraic & Geometric Topology

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

#### 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 $cat1(X)$ (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

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

Dates
Revised: 22 April 2010
Accepted: 24 April 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715129

Digital Object Identifier
doi:10.2140/agt.2010.10.1165

Mathematical Reviews number (MathSciNet)
MR2653059

Zentralblatt MATH identifier
1204.55004

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715129

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