Absolute Whitehead torsion

We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category \bA from an element of \widetilde{K}^{iso}_1(\bA) to an element of the absolute group K_1^{iso}(\bA). We apply this invariant to symmetric Poincar\'e complexes and identify it in terms of more traditional invariants. In the companion paper [1] (joint with Ian Hambleton and Andrew Ranicki) this new invariant is applied to obtain the multiplicativity of the signature of fibre bundles mod 4.Comment: To appear in the MPI preprint serie

Noncommutative localization in algebraic LL-theory

Given a noncommutative (Cohn) localization $A \to \sigma^{-1}A$ which is injective and stably flat we obtain a lifting theorem for induced f.g. projective $\sigma^{-1}A$-module chain complexes and localization exact sequences in algebraic $L$-theory, matching the algebraic $K$-theory localization exact sequence of Neeman and Ranicki.Comment: to appear in Advances in Mathematic

Foundations of algebraic surgery

An elementary introduction to the principles of algebraic surgery.Comment: 19 pages. Notes of lecture given at the Summer School on High-dimensional Manifold Topology, ICTP Trieste, May-June 2001. To appear in Vol. 1 of the Proceeding

The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction

An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.Comment: 20 pages. Notes of lecture given at the Summer School on High-dimensional Manifold Topology, ICTP Trieste, May-June 2001. To appear in Vol. 1 of the Proceeding

Algebraic Poincare cobordism

This paper is an introduction to the use of the cobordism of chain complexes with Poincar\'e duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of Surveys in Surgery Theory, Ann. of Maths. Studies, Princeton, 2001) which is an introduction to algebraic surgery using forms and formations.Comment: AMSTEX 43 pages, conm-p.sty, ams-p.sty, ams-spec.sty and xypic input to appear in Proc. 1999 Stanford conference in honour of 60th birthday of R.J.Milgra

Blanchfield and Seifert algebra in high dimensional knot theory

Novikov initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relationship between the Seifert forms over a ring with involution and Blanchfield forms over the Laurent polynomial extension.Comment: 30 pages, LATEX. v3: minor revision of v2 (which was itself a minor revision of v1

Noncommutative localization in topology

A survey of the applications of the noncommutative Cohn localization of rings to the topology of manifolds with infinite fundamental group, with particular emphasis on the algebraic K- and L-theory of generalized free products.Comment: 20 pages, LATEX. To appear in the Proceedings of the Conference on Noncommutative Localization in Algebra and Topology, ICMS, Edinburgh, 29-30 April, 2002. v2 is a minor revision of v

Circle valued Morse theory and Novikov homology

An introduction to circle valued Morse theory and Novikov homology, from an algebraic point of view.Comment: 26 pages. Notes of lecture given at the Summer School on High-dimensional Manifold Topology, ICTP Trieste, May-June 2001. To appear in Vol. 1 of the Proceeding

A second order algebraic knot concordance group

We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single invariant. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental groups, and in particular on the way in which they can change in a concordance.Comment: 51 pages, 2 figures. This a considerably shortened version of arXiv:1109.0761, to appear in Algebraic and Geometric Topolog