In this paper, we study the growth with respect to dimension of quite general homotopy
invariants
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
–category. We
use (which is
–category with
the collection
of –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
,
which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom.
Topol. 10 (2010) 917–924].
Keywords
Lusternik–Schnirelmann category, skeleta, fundamental
group, symplectic manifold
