Signature operators and surgery groups over \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Signature operators and surgery groups over \(C^*\)-algebras
zbMATH Open document ID: 0920.19002
DOI: 10.1023/A:1007768122610
author: John G. Miller
published in: \(K\)-Theory
publication date: 1999-09-15T00:00:00Z
review text: Let \(A\) be a unital complex \(C^*\)-algebra, the author defines a group \({\mathcal B}^*(A)\) of bordism classes of Fredholm complexes over \(A\) with Poincaré duality. They generalize the de Rham complex of a manifold. These fit into a natural commutative triangle \[ \begin{matrix} {\mathcal B}^*(A) & @>S>> & K_*(A)\\ \sigma \downarrow & & \downarrow =\\ L^*(A) & @>>m> & K_*(A) \end{matrix} \] where \(L^*(A)\) denotes the symmetric surgery groups associated to an involutive algebra by Ranicki. The map \(\sigma\) is given by the ``algebraic signature map''. It is given on complexes by the forgetful functor to the associated Witt group and descends to a map of bordism groups. The map \(S\) is the analytic signature which is given by the index of the ``signature operator'' associated to a Fredholm-Poincare complex. Finally \(m\) assigns to a quadratic form over a \(C^*\)-algebra its signature, i.e., the formal difference of the spectral projections onto the positive and negative parts of the spectrum of the operator implementing the quadratic form. The reader should note that this description makes sense only in even degrees and that the established triangle contains new information in odd degrees. The author proves that all maps in the triangle are isomorphisms of abelian groups.
reviewed by: Michael Puschnigg
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J10
zbMATH DE Number: 1139537
zbMATH Keywords: signature operator
zbMATH Keywords: Poincaré complex
zbMATH Keywords: symmetric signature
zbMATH Keywords: algebraic bordism
zbMATH Keywords: Fredholm complexes
zbMATH Keywords: surgery groups
MaRDI profile type: Publication
full work available at URL: a:1007768122610
OpenAlex ID: W2063994416
description: scientific article; zbMATH DE number 1139537