2 JOHN W. RUTTER CONTENTS 1. Surfaces, manifolds 2. Localization 3. £*(X) finitely presented, nilpotent 4. Methods for calculation 5. Calculations made 6. Structural properties 7. Homotopy type, homotopy groups 8. Fibre and equivariant homotopy equivalences 9. Applications 1. Surfaces, manifolds 1.1. Surfaces. Let T 9 be a compact orientable surface of genus g. Let G be a subgroup of £(T9 ), then either G contains an abelian subgroup of finite index or G contains a non-abelian free group (see Ivanov [1992, Theorem 4]). £(T9 ) does not contain an arithmetic subgroup of finite index for g ::: 2 (Ivanov [1988B]). The Ballman-Eberlien rank of £(T9 ) is 1 for for g ::: 1 (Ivanov [1988A]). A presentation for the symplectic modular group Sp+(2g, Z) (g ::: 3) on 3 generators and 3g + 5 relators was given by Lu [1992, Theorem 3]. 1.2. Higher ~anifolds. For orientable geometric 3-manifolds, the group of simple homotopy self-equivalence classes S(M) lies in the image of 1i(M) ----t £(M) (Turaev [1988], see also Kwasik and Schultz [1992, Theorem 1.1]). In the case of a compact connected orientable 3-manifold M = Ml# ... #Mn#(#k(S1 X 8 2 )), McCullough [1990] and Hatcher and McCullough [1990] obtained results on finite generation and finite presentation of the homeotopy group 1i(M) and of K(M), the (orientation-preserving subgroup of the) kernel of 1i(M) ----t outrr1(M). Mc- Cullough [1991, Theorem page 2] and Kalliongis and McCullough [1992, Theorem 4.2.3] showed that 1i(M) is finitely presented in case M is a compact connected irreducible sufficiently large (in the sense of Swarup in the non-orientable case) 3-manifold. Ltc. X be a compact connected 4-manifold with finite-dimensional fundamental group and f : X ----t B where B is the second Postnikov stage of X. Hayat and Legrand [1996] obtained several exact sequences including H 1(rr1, rr2) ----t £B(X) ----t £(X) ----t £(B)f ----t 1 H 2(B,rr3) ----t Ho(rr~,rr4) ----t £B(X) ----t HI('Il'I,11'3)· In case rr1 has periodic cohomology of period 4, they calculated £(X) algebraically, up to extension. For a closed, compact, oriented manifold M, Baues [1996] studied the sequence 0 ----t £A(M) ----t £ ----t E(A, 8 1) ----t 1 where A= M\eM, 8 1 is the boundary of A, £A(M) is the image of£A(M) ----t l(M) and E(A, 8 1) is the image of £(A, 8 1) ----t £(A). The group £A(M) is finite and
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