Concordance Spaces, Higher Simple- Homotopy Theory, and Applications

While much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. In fact, it seems that there is not a single closed manifold M of dimension greater than three for which the homotopy type of the automorphism space Diff(M), PL(M), or TOP(M) in the smooth, PL, or topological category, respectively, is in any sense known. (As usual, Diff(M) is given the C°° topology, PL(M) is a simplicial group, and TOP(M) is the singular complex of the homeomorphism group with the compact-open topology.) Besides surgery theory, the principal tool in studying homotopy properties of these automorphism spaces is the concordance space functor C(M) = {automorphisms of M x /fixed on M x 0}. This paper is a survey of some of the main results to date on concordance spaces. Here is an outline of the contents. In §1 we describe how, in a certain stable dimension range, C{M) is a homotopy functor of M, which we denote by ^(M). The application to automorphism spaces is outlined in §2. In §3 we recall the explicit calculations which have been made for %$?(M) and %{g(M), along the lines pioneered by Cerf, and apply them in §4 to compute the group of isotopy classes of automorphisms of the «-torus, n ^ 5. §5 is concerned with a stabilized version of #(Af), defined roughly as Q^iS^M), together with the curious equivalence of D ^ P L O S ^ M ) with ^PL(M)/^Dif f(M), due to Burghelea-Lashof (based on earlier fundamental work of Morlet). In §6, ^P L(M) is "reduced" to higher simplehomotopy theory. This has some interest in its own right, e.g., it provides a fibered form of Wall's obstruction to finiteness. The important new work of Waldhausen relating <^PL(M) to algebraic AT-theory is outlined, very briefly and imperfectly, in §7. This seems to be the most promising area for future developments in the sub-