4 JOHN W. RUTTER Roitberg [1991] proved that £*(X)~ R)q (Z2 x Z2 ) is solvable and uncountable but not residually nilpotent for the simply-connected space X= K(Z, 2) X S 3 . D.W. Kahn [1990A], complementing his earlier result, also showed, essentially, for a finite connected stable complex, that the image of the stable homotopy ( fac- tored by torsion) representation £*(X) ----+ I1 aut 1r~ (X) F has finite index. Triantafillou [1992, Theorem 1.2 and Corollary 1.5], extending the Sullivan- Wilkerson result, showed that stabt(f(X)) is finitely presented and of finite type where X is a finite nilpotent CW-space, Y is a nilpotent CW-space of finite type, and f E (X, Y)*. This was further extended by Maruyama [1996], who showed that the subgroups of £*(X), £*(Y) and £*(X) x £*(X) which leaveS (pointwise) invariant are all finitely presented, where X and Y are finite nilpotent complexes and Sis a finite subset of (X, Y)* also any nilpotent subgroup of £*(X) is finitely presented. Triantafillou also showed [1995] that £(X) is finitely presented provided that X is a finite complex with finite fundamental group, the group of homo- topy classes of tangential homotopy self-equivalences (also for tangential simple homotopy self-equivalences) of an oriented manifold with finite fundamental group is finitely presented, and the group of homotopy classes of diffeomorphisms of a smooth closed oriented manifold of dimension ~ 5 with finite fundamental group is finitely presented. The results in general show that the appropriate group is commensurable with an arithmetic group. By the result of Formanek and Procesi [1992], we have that the free group £*(X) is not linear in the case where X is a bouquet of n circles (n ~ 3), though it is finitely presented. Arkowitz and Lupton [1995, Appendix] showed that KH-(X) and all its sub- groups are finitely generated, where X is a !-connected finite complex. 4. Methods for calculation 4.1. Cell complexes. Rutter [1988A, Theorem 1 and Theorem 2] gave meth- ods for calculating£* (X), in case X is simply-connected, by moving progressively up the stages of a homology decomposition. Let f(X) be the subgroup of £(X) consisting of those classes which can be represented by a cellular map f : X ----. X which is a homotopy self-equivalence r : xr ----. xr on each skeleton xr of X. Rutter [1990A, Theorem B] gave conditions (e.g. X is simply-connected finite complex with X= {pt}) which ensure that f(X) has finite index in £(X). Kahn [1992] discussed the twisted torus and characteristic classes of a homotopy self-equivalence. 4.2. Non-1-connected Postnikov pieces. Methods were given by Rutter [1992B, Theorem 2.4, Theorem 2.9] for calculating £*(X) and £(X) for connected spaces by proceeding up the stages of a Postnikov decomposition. In the case of a non-simply connected space having precisely two non-trivial homotopy groups and k-invariant h, we have the group extension (obtained independently by M0ller [1991, Theorem 5.6]) 0---- H (Kl(7Tl) 7Tn)----+ £*(X)----+ stabh(aut1r1 Xq, aut7rn)---- 1, where ¢ is the action of 1r1 on 1Tn· In the special case where X = L~(1Tn) is the generalized Eilenberg-McLane space, the sequence splits (Rutter [1992B, Lemma 2.1
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