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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The two definitions of the index difference
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by Johannes Ebert PDF
Trans. Amer. Math. Soc. 369 (2017), 7469-7507 Request permission

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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Additional Information
  • Johannes Ebert
  • Affiliation: Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Bundesrepublik Deutschland
  • MR Author ID: 811149
  • Email: johannes.ebert@uni-muenster.de
  • 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
  • © Copyright 2017 American Mathematical Society
  • 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
  • MathSciNet review: 3683115