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The two definitions of the index difference


Author: Johannes Ebert
Journal: Trans. Amer. Math. Soc. 369 (2017), 7469-7507
MSC (2010): Primary 19K56, 53C21, 53C27, 55N15, 58J30, 58J40
DOI: https://doi.org/10.1090/tran/7133
Published electronically: June 13, 2017
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Abstract: Given two metrics of positive scalar curvature on a closed spin manifold, there is a secondary index invariant in real $ K$-theory. There exist two definitions of this invariant: one of a homotopical flavor, the other one defined by an 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 result to the case of two families of positive scalar curvature metrics, parametrized by a finite CW complex. 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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Johannes Ebert
Affiliation: Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Bundesrepublik Deutschland
Email: johannes.ebert@uni-muenster.de

DOI: https://doi.org/10.1090/tran/7133
Keywords: Fredholm model for $K$-theory, Bott periodicity, spectral flow, Dirac operator, positive scalar curvature
Received by editor(s): August 22, 2013
Received by editor(s) in revised form: October 7, 2014, August 7, 2015, August 10, 2015, April 12, 2016, and November 17, 2016
Published electronically: June 13, 2017
Article copyright: © Copyright 2017 American Mathematical Society