Homotopical Excision, and Hurewicz Theorems, for n ‐Cubes of Spaces

where e is induced by the ‘excision’ inclusion. That e is an isomorphism in a range of dimensions is shown by the classical Blakers-Massey triad connectivity theorem: if A, B, and A∩B are connected, {A,B} is an open cover of X and (A, A ∩ B) is p-connected, (B, A ∩ B) is q-connected, then the triad (X; A,B) is (p + q)-connected. (See, for example, [11, p. 211].) Further, if p, q > 2 and π1(A∩B) = 0, the critical group πp+q+1(X; A, B) is described in [2] as a tensor product of abelian groups πp+1(A, A ∩ B)⊗ πq+1(B, A ∩ B). One of our main results (Theorem 4.2) extends this description of the critical group to the cases where p, q > 1, π1(A ∩ B) = 1 = 0. Note that if p or q is 1, then one at least of the groups πp+1(A, A∩B), πq+1(B, A∩B) may be non-abelian, and acts on the other group. In the description of the critical group, the usual tensor product must be replaced by the tensor product G⊗H defined in [5, 6], which involves actions of G on H and H on G. This description of πp+q+1(X;A, B) is a special case of a description of the hyper-relative group πn+1(X; A1, . . . , An) of a ‘connected’ excisive (n+1)-ad as determined by the lower dimensional information involved in the (n+1)-ad; a precise description is given in Theorem 4.1. As another consequence of Theorem 4.1 we obtain an exact sequence for a connected space