Diffeomorphisms of Manifolds with Finite Fundamental Group

We show that the group 3i{M) of pseudoisotopy classes of diffeomorphisms of a manifold of dimension > 5 and of finite fundamental group is commensurable to an arithmetic group. As a result n0{DiffM) is a group of finite type. Let M be an «-dimensional closed smooth manifold, where n > 5, and let DiffM be the group of diffeomorphisms of M. The space DiffM (it is a topological space with the C°°-topology) is in general very complicated and has been studied by many authors. Its homotopy type is known only in some special cases. We recall the well-known results Diff(S2) ~ 0(3) by Smale [Sm] and Diff(S3) ~ 0(A) by Hatcher [H]. The component group no(DiffAT) has also been initially computed in special cases, for instance for spheres, where the group nç>(Diff+Sn) of orientationpreserving diffeomorphisms is isomorphic to the group of homotopy spheres of dimension n + 1 for n > A [KM], and for products of spheres [B, Tu]. The first general result about the nature of n0(DiffM) for simply connected manifolds M is due to Sullivan [S], who showed that if M is a smooth closed orientable simply connected manifold of dimension > 5, then no(DiffM) is commensurable to an arithmetic group. Arithmetic groups and their properties have been studied in [BH] by Borel and Harish-Chandra. In particular they showed that every arithmetic group is finitely presented. It follows that no(DiffM) is finitely presented if M is simply connected. In fact no(DiffM) in this case is a group of finite type by a result of Borel and Serre that arithmetic groups are of finite type [BS]. By definition a group n is of finite type if its classifying space Bn is homotopy equivalent to a CW-complex with finitely many cells in each dimension. Being of finite type for a group implies and in fact is much stronger than finite presentation. Two groups are said to be commensurable if there is a finite sequence of homomorphisms between them which have finite kernels and images of finite index. On the other hand na(DiffM) can be very large if M is not simply connected. For instance, Tto(DiffT") contains a subgroup isomorphic to a direct sum of infinitely many copies of Z2 for n > 5 , and 7to(Dijf(Tn x S2)) contains a free abelian subgroup of infinite rank for « > 3, where T" is the torus [HS]. In these cases ni(M) is infinite. In this paper we study no(DiffM) and 3i(M) in the case where ii\(M) is Received by the editors September 30, 1993. 1991 Mathematics Subject Classification. Primary 57R50, 57R52, 57S05; Secondary 57R67, 55P62.