Higher Index Theorems and the Boundary Map in Cyclic Cohomology

We show that the Chern--Connes character induces a natural transformation from the six term exact sequence in (lower) algebraic $K$--Theory to the periodic cyclic homology exact sequence obtained by Cuntz and Quillen, and we argue that this amounts to a general ``higher index theorem.'' In order to compute the boundary map of the periodic cyclic cohomology exact sequence, we show that it satisfies properties similar to the properties satisfied by the boundary map of the singular cohomology long exact sequence. As an application, we obtain a new proof of the Connes--Moscovici index theorem for coverings.

1991 Mathematics Subject Classification: (Primary) 19K56, (Secondary) 19D55, 46L80, 58G12.

Key Words: cyclic cohomology, algebraic $K$-theory, index morphism, etale groupoid, higher index theorem.

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