By Matthias Lesch
The authors exhibit the Connes-Chern of the Dirac operator linked to a b-metric on a manifold with boundary when it comes to a retracted cocycle in relative cyclic cohomology, whose expression depends upon a scaling/cut-off parameter. Blowing-up the metric one recovers the pair of attribute currents that signify the corresponding de Rham relative homology category, whereas the blow-down yields a relative cocycle whose expression contains better eta cochains and their b-analogues. The corresponding pairing formulae, with relative K-theory sessions, catch information regarding the boundary and make allowance to derive geometric outcomes. As a spinoff, the authors convey that the generalized Atiyah-Patodi-Singer pairing brought through Getzler and Wu is unavoidably limited to just about flat bundles.
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