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 [Show abstract] [Hide abstract] ABSTRACT: This chapter presents the definition of the generalized Whitehead products in an (n+1)ad and discusses the first nonvanishing homotopy group of a complete CW(n–1)ad in terms of these products. This expression involves nfold products. Thus, the first nonvanishing group of a tetrad is described by means of triple products, of a 5ad by quadruple products and so on. The chapter discusses the conditions under which the product homomorphism θ: πm(A, C) ⊗ πq–m+1 (B, C) → πq(X,A, B) for a triad is an isomorphism onto, or merely onto, when q takes on a range of values beyond the first for which πq(X, A, B) ≠ 0.


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 [Show abstract] [Hide abstract] ABSTRACT: Let G be a tower of finitely generated nilpotent groups and let G(p) denote its localization at a prime p. It is shown that if lim1 (G) is nontrivial, then it contains at least two elements whose images are equal in lim1 (G(p)) for every prime p. Are there, in fact, infinitely many elements with this property? For abelian towers, the answer is yes. It is shown that the answer is also yes for certain nonabelian towers. Other questions considered in this paper include the problem of characterizing the lim1 term of towers of countable abelian groups, towers whose lim1 terms are countably infinite, and the extent to which the lim1 term depends upon the actual group structures in the tower.
 [Show abstract] [Hide abstract] ABSTRACT: Let A be a fixed collection of spaces, and suppose K is a nilpotent space that can be built from spaces in A by a succession of cofiber sequences. We show that, under mild conditions on the collection A, it is possible to construct K from spaces in A using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if K is a nilpotent finite complex, then OmegaK can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied to show that if map(*) (X, S(n)) is weakly contractible for all sufficiently large n, then map(*) (X, K) is weakly contractible for any nilpotent finite complex K.