The algebra of cubes∗†

X∗ : X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X. We replace the fundamental group π1(X, ∗) by the homotopy crossed complex πX∗ which consists of the family of groups Cn(p) = πn(Xn, Xn−1, p) for n > 2 and all p ∈ X0, together with the fundamental groupoid C1 = π1(X1, X0) over C0 = X0, all with the standard boundary maps Cn(p) → Cn−1(p) and action of C1. ∗This research was supported by the Science Research Council Grants B/RG/5736.8, GR/A/3484.9, GR/A/0957.1. †This is a version with minor revisions and in LTEX of the paper published in J. Pure Applied Algebra 21 (1981) 223-260