The Homotopy Groups of a Triad. III

The principal purpose of this paper is to prove a rather general theorem about the homotopy groups of a triad in what may be called the "critical dimension," i.e., the lowest dimension for which the homotopy groups of a triad are non-zero. This theorem may be stated roughly as follows. Let (X j A, B) be a triad such that X = A u B. If 1I"1'(A, A n B) = 0 for p ~ m and 1I"q(B, A n B) = 0 for q ~ n, then the authors have shown previously [2] that ·lIAXj A, B) = 0 for r ~ m + n under very general conditions. We now show that 1I"m+>l.H(X; A, B) is isomorphic to the tensor product, 1I"mH(A, A n B) ® 1I">l.+,(B, A n B), under rather general conditions. Moreover, this isomorphism is defined in a very natural manner by means of a generalized Whitehead product. This theorem includes as special cases some results we have announced previously without prooe The proof which we give below depends heavily on a recent paper of J. C. Moore, [81. This proof is much simpler than the authors' original, unpublished proofs for the previously announced results. In sections 2 and 3 we give some applications of our main theorem to some problems of current interest in algebraic topology. This paper is essentially a continuation of our earlier papers, [1], [2], and [31. For the explanation of any terminology or notation that is not contained in the present paper, the reader is referred to these previous papers. In general, it is assumed that the reader is familiar with the basic properties of triad homotopy groups and generalized Whitehead products.