Advertisement

Automorphism L-theory

  • Andrew Ranicki
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

This chapter deals with the L-theory of automorphisms of algebraic Poincaré complexes, using the localization Ω −1 A[z, z −1] inverting the set Ω (13.3) of Fredholm matrices in the Laurent polynomial extension \(A\left[ {z,{z^{ - 1}}} \right]\left( {\bar z = {z^{ - 1}}} \right)\) of a ring with involution A. In the first instance, the duality involution on W h 1(Ω −1 A[z, z −1]) and the fibering obstruction are used to prove that every manifold band is cobordant to a fibre bundle over S 1, allowing the identification of the closed framed spine bordism groups Δ *(X, F) of Chap. 27 for
$$\mathcal{F} = inclusion:\mathbb{Z}\left[ {{{\pi }_{1}}\left( X \right)} \right]\left[ {z,{{z}^{{ - 1}}}} \right] \to {{\Omega }^{{ - 1}}}\mathbb{Z}\left[ {{{\pi }_{1}}\left( X \right)} \right]\left[ {z,{{z}^{{ - 1}}}} \right]$$
with the bordism groups Δ *(X) of automorphisms of manifolds over a space X.

Keywords

Exact Sequence Fibre Bundle Invariant Subgroup Homotopy Equivalent Mapping Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Andrew Ranicki
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of EdinburghEdinburghScotland, UK

Personalised recommendations