The two definitions of the index difference

• Published in: Transactions of the American Mathematical Society Vol. 369; no. 10; pp. 7469 - 7507
• Main Author: EBERT, JOHANNES
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-10-2017
• Subjects: spectral flow; positive scalar curvature; Fredholm model for $K$-theory; Bott periodicity; Dirac operator
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7133

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.