Automorphisms of Manifolds and Algebraic K-Theory Finale, by M. S. Weiss and Bruce Williams

Let M be a closed topological n-manifold, and let S(M) be the moduli space of closed topological manifolds equipped with a homotopy equivalence to M. We give an algebraic description of S(M) in the h-cobordism stable range, assuming n>4. (That is, we produce a highly connected map from S(M) to another space having an algebraic description.) The algebraic description is in terms of L-theory, Waldhausen's algebraic K-theory of spaces, and a natural transformation (constructed in an earlier paper by us) from L-theory to the Tate cohomology of Z/2 acting on K-theory.

We develop a parallel theory for the moduli space S(tangent bundle) of R^n-bundles on M equipped with an "equivalence" to the tangent bundle of M. (The equivalence is a stable fiber homotopy equivalence of the corresponding spherical fibrations.) Results about moduli spaces of smooth manifolds can be obtained by combining the calculations of S(M) and S(tangent bundle).

We have attempted to make this paper as self-contained as possible by summarizing results from the earlier papers in the series where necessary.

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M. S. Weiss <msweiss@math.lsa.umich.edu>
Bruce Williams <bruce@bruce.math.nd.edu>