The Homotopy Groups of a Triad.

induce isomorphisms of the relative homology groups in all dimensions. Simple examples show that this need not be true for the relative homotopy groups, even when X is a finite connected simplicial complex, and A, B, and A n B are connected subcomplexes. The new homotopy groups defined in this paper are a measure of the amount by which the excision axiom fails to hold for relative homotopy groups. The precise meaning of this statement will be clear later. One special case for which it is particularly important to determine the extent of the validity of the excision axiom for relative homotopy groups is the following: Let K be a cell complex2 and let K', n 0, 1, 2, * * , denote the n-skeleton. Denote the closed n-cells of K by al', a, .* , their boundaries by al , &n, **, and set