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

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 [7].