Publisher Summary This chapter discusses second non-vanishing homotopy groups of pairs and triads. The ( r + 1)th homotopy group, π r +1 ( Y ), of a finite, ( r –1)-connected closure finite complexes with weak topology (CW-complex) Y , where r ≥ 3, has been calculated in terms of the homology theory of Y . The chapter describes Ext( Q , G ), when pG = 0 for any prime p , and apply the results presented in the chapter to the computation of π r+1 ( Y ), when Y is an arbitrary ( r –1)-connected CW-complex. If L is a 2-connected sub-complex of a CW-complex if, ( K , L ) is ( r –1)-connected, then the group π r +1 ( K , L ) can be found by pinching L to a point. The chapter presents the structure of π r +1 ( K, L ) when K = A× B , L = A B , in terms of the homology groups of A , B .