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In this paper, we introduce an alternative definition of the Chang--Weinberger--Yu relative higher index as a relative analogue of the Mishchenko--Fomenko index pairing. We use this definition to construct the dual relative higher index map, which is related to the index pairing with almost flat bundles.

The relative Mishchenko–Fomenko higher index and almost flat bundles. I. The relative Mishchenko–Fomenko index