Automorphism L-theory

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


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.


Exact Sequence Fibre Bundle Invariant Subgroup Homotopy Equivalent Mapping Torus 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

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

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