CHAPTER 1 Introduction The structure space S(M) of a closed topological m-manifold M is the clas- sifying space for bundles E X with an arbitrary CW -space X as base, closed topological manifolds as fibers and with a fiber homotopy trivialization E M × X (a homotopy equivalence and a map over X). The points of S(M) can loosely be imagined as pairs (N, f) where N is a closed m-manifold and f : N M is a homo- topy equivalence. To explain the relationship between S(M) and automorphisms of M, we invoke Hom(M), the topological group of homeomorphisms from M to M, and G(M), the grouplike topological monoid of homotopy equivalences from M to M. In practice we work with simplicial models of Hom(M) and G(M). The homotopy fiber of the inclusion BHom(M) BG(M) is homotopy equivalent to a union of connected components of S(M). The main result of this paper is a calculation of the homotopy type of S(M) in the so-called concordance stable range, in terms of L- and algebraic K-theory. With m fixed as above, we construct a homotopy invariant functor (Y, ξ) LA•%(Y, ξ, m) from spaces Y with spherical fibrations ξ to spectra. The spectrum LA•%(Y, ξ, m) is a concoction of the L-theory and the algebraic K-theory of spaces [27] associated with Y , compounded with an assembly construction [21]. (The subscript % is for homotopy fibers of assembly maps.) In the case where Y = M (nonempty and connected for simplicity) and ξ is ν, the normal fibration of M, there is a “local degree” map Ω∞+mLA•%(M, ν, m) −→ 8Z Z. There is then a highly connected map (1.1) S(M) −→ fiber Ω∞+mLA•%(M, ν, m) local deg. −−−−−− 8Z where fiber in this case means the fiber over 0 8Z, an infinite loop space. The connectivity estimate is given by the concordance stable range. In practice that translates into m/3 approximately, but in theory it is more convoluted and the reader is referred to definition 11.5. The result has a generalization to the case in which M is compact with nonempty boundary. It looks formally the same. Points of S(M) can be imagined as pairs (N, f) where N is a compact manifold with boundary and f : (N, ∂N) (M, ∂M) is a homotopy equivalence of pairs restricting to a homeomorphism of ∂N with ∂M. We now give a slightly more detailed, although still sketchy, definition of the spectrum ΩmLA•%(Y, ξ, m). (Details can be found in chapter 9.) It is the total 1
Previous Page Next Page