Bordism of automorphisms from the algebraic L-theory point of view, by Andrew Ranicki

The mapping torus construction defines a natural transformation T from the bordism over a space X of n-dimensional manifolds M with an automorphism f:M-->M to the bordism of (n+1)-dimensional manifolds N over XxS^1. Quinn used open book surgery to identify the fibre of T (in dimensions n>5) with the asymmetric Witt theory of the fundamental group ring Z[\pi_1(X)]. In this paper the fibre of T is identified with the quadratic Wall L-theory of the localization \Omega^{-1}Z[\pi_1(XxS^1)] inverting the set \Omega of square matrices in the Laurent polynomial ring Z[\pi_1(XxS^1)]=Z[\pi_1(X)][z,z^{-1}] which present f.g. projective Z[\pi_1(X)]-modules. The essential ingredient is the observation that the canonical infinite cyclic cover of a finite CW complex N with \pi_1(N)=\pi_1(XxS^1) is finitely dominated if and only if H_*(N;\Omega^{-1}Z[\pi_1(XxS^1)])=0.


Andrew Ranicki <a.ranicki@edinburgh.ac.uk>