THE FIBER OF AN ITERATIVE ADJUNCTION OF CELLS

Let (Y,X) be a relative CW complex with X and Y simply-connected and s uppose that the relative homology H(Y,X;k) is nonzero. Denote by F th e homotopy fibre of the inclusion X hooked right arrow Y. We show that the grade of H(F;k) as a module over H*(Omega Y;k) is less than the relative cone length cl(Y,X). This result appears as a corollary of a deeper result concerning differential modules over differential graded algebras. (C) 1997 Elsevier Science B.V.