The fibre of an iterative adjunction of cells Original Research Article

Let (Y,X) be a relative CW complex with X and Y simply-connected and suppose that the relative homology H*(Y, X; k) is nonzero. Denote by F the homotopy fibre of the inclusion X → Y. We show that the grade of H*(F;k) as a module over H*(ΩY;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.