Coarse Homotopy - Trung's Thesis

This is a small advertising memo about part of Viet-Trung Luu's forthcoming thesis, which contains a very elegant unified treatment of various theorems about the 'coarse homotopy' or 'Lipschitz homotopy' invariance of the K-theory of C*(X). It formalizes the basic idea of the paper by Nigel and me in the Transactions, found here. One needs "C*(X) with coefficients", defined using Hilbert modules. Trung's approach has a geometric and an analytic part. The geometric part is to show that a coarse homotopy gives an element of C*(X; C[0,1]). The analytic part is to show that there is a natural pairing between the K-theory of C*(X;D) and the K-homology of D, with values in the K-theory of C*(X). Now one uses the known homotopy invariance of K-homology - the fact that the inclusions of 0 and 1 into [0,1] induce isomorphisms on K-homology - to conclude the coarse homotopy invariance of K_*(C*(X)).