The category of a map and the grade of a module

Abstract

Let f : Y --> X be a continuous map between connected CW complexes. The homology H* (F) of the homotopy fibre is then a module over the loop space homology H* (OMEGAX). THEOREM: If H*(F; R) and H*(OMEGAX; R) are R-free (R a principal ideal domain) then for some H*(OMEGAX; R)-projective module P=P(greater-than-or-equal-to 0) and for some m less-than-or-equal-to cat f: Ext(H* (OMEGAX)m(H* (F); P) not-equal 0. Some applications are also given.