HOMOTOPY SELF-EQUIVALENCES 1988-1999 3 abelian special results hold if M is 2n--dimensional and ( n - 1 )-connected, or is ( n - 1 )-connected where m 3n - 1. Where n 11 and M is a connected sum of copies of 8m x 8", £(M) was calculated in many cases. Baues and Buth [1996] made substantial calculations of £*(M) for simply-connected 5-manifolds. Let G be a compact connected Lie group, then £(BG) ~ out G, the group of Lie group outer automorphisms (Jackowski, McClure and Oliver [1995] and Osse [1997]) also the finite homology type of E(BG, 1) was determined as BZ where Z is the centre of G (Jackowski, McClure and Oliver [1992A] and [1992B]). The ~-local homotopy type of E(M)/ H(M), for a compact topological man- ifold M, is the loop space of the Hermitian K -theory Whitehead space of M (Fiedorowicz, Schwiindl and Vogt [1992]). In the PL--category, for n ~ 2, the mapping class group M(#p(81 x 8")) ~ £(#p(81 x 8")) 0 r, the group of orientation-preserving homotopy self-equivalence classes, and there is the exact sequence 0-+ ffipZ2-+ £(#p(81 X 8")) 0 r-+ outFp -+1 (Cavicchioli, Hegenbarth and Spaggiari [1988]). 2. Localization K~(X)-+ K~(Xp) P-localizes in case X is a connected simple finite dimen- sional CW--complex of finite type, and K'J}(X) -+ K'J}(Xp) P-localizes in case X is a simply-connected finite CW--complex (Maruyama ([1989, Theorem 0.1] and [1990A, page 291])). Here P is a set of primes, K~(X) is the kernel of the repre- sentation £(X) -+ f} _" aut 11'r{X), and K'J}(X) is the kernel of the representation £(X) -+ n _n aut Hr(X). Calculations of KH(X)p and K1r:(X)p for a number of H- spaces were given by Maruyama ([1990A, Example 3.1] and [1990B, 3.1]). Results on localization were also given by M!llller ([1992, Corollary 3.2] and [1989, Theorem 4.3]). See also §8 of Rutter [1997]. Arkowitz [2] showed that for a minimal DGA A, £(A)~ autH*(A) l Kc(A) and give an algorithm for calculating K1r:(A)). Let £o(X) C £(X) consist of the classes of rational equivalences, and let X be a space whose p-completion (X)~ has the homotopy type of (BG)~ for any prime p, where G = 8 3 x · · · x 8 3 • Ishiguro, M!llller and Notbohm [1999] studied £0 (X) a submonoid of £o(X) determines the decomposability of X. 3. £*(X) finitely presented, nilpotent Dror and Zabrodsky had shown that KH- is a nilpotent group in case X is a path-connected nilpotent finite dimensional CW -complex. Maruyama and Mimura [1990] extended this to suitable generalised homology theories they showed also that it is not true forK-theory. Xu [1996] showed that if a subgroup G of £*(X) acts on Ei(X) nilpotently then G is nilpotent, where X is finite connected, nilpotent and has only p-torsion homology, and E is a connective spectrum. Felix and Murillo [1997] showed that the nilpotency of the rationalization of £*(X)) is ::: cat(X) - 1 and obtained a corresponding result for fibre homotopy equivalences in [1998] they showed that, if cat (X) is finite, the kernel of £*(X) -+ £*(S1X) is nilpotent and they give a bound for the nilpotency class. Arkowitz and Lupton [1996] studied the nilpotency class of K (X) and obtain bounds for it for certain finite complexes.

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