A twisted free tensor product of a differential algebra and a free differential algebra is introduced. This complex is proved to be chain homotopy equivalent to the complex associated with a twisted free product of a simplicial group and a free simplicial group. In this way we turn a geometric situation into an algebraic one, i.e. for the cofibration Y → Y ∪ g C X → Σ X Y \to Y\,{ \cup _g}\,CX \to \Sigma X we obtain a spectral sequence converging into H ( Ω ( Y ∪ g C X ) ) H(\Omega (Y\,{ \cup _g}\,CX)) . The spectral sequence obtained in the above situation is similar to the one obtained by L. Smith for a cofibration. However, the one we obtain has more information in the sense that differentials can be traced, requires more lax connectivity conditions and does not need the ring of coefficients to be a field.