Signature Operators and Surgery Groups over C*-algebras

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Abstract

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 Miščenko.

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