ON THE SECOND NON-VANISHING HOMOTOPY GROUPS OF PAIRS AND TRIADS

Abstract

This chapter discusses second non-vanishing homotopy groups of pairs and triads. The (r + 1)th homotopy group, πr+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 πr+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 πr+1 (K, L) can be found by pinching L to a point. The chapter presents the structure of πr+1 (K, L) when K = A× B, L = A B, in terms of the homology groups of A, B.