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 $\mathcal{Q}$ 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 $\mathcal{A}$–category. We use ${cat}^1\left(X\right)$ (which is $\mathcal{A}$–category with $\mathcal{A}$ 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].