The two definitions of the index difference

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this to the case of two families of positive scalar curvature metrics, parametrized by a compact space. In essence, we prove a generalization of the classical "spectral-flow-index theorem" to the case of families of real operators.

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