Signature operators and surgery groups over \(C^*\)-algebras
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.
scientific article; zbMATH DE number 1139537