Signature Operators and Surgery Groups over C*-algebras

Let A be a unital complex C^* algebra, L^*(A) the projective symmetric surgery groups, and K_*(A) topological K theory. We define groups B^*(A) of bordism classes of Fredholm complexes over A with Poincaré duality. These generalize the de Rham complex. It is shown that there are isomorphisms B^*(A)K_* (A) and B^*(A) L_*(A) given by abstract versions of the signature operator and symmetric signature. The remaining side of a triangle is formed by an isomorphism due to Mienko.