Families of Dirac operators, boundaries and the $b$-calculus

Abstract

A v ersion of the Atiyah-Patodi-Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary. T h e v anishing of the analytic index of the boundary family, i n K 1 of the base, allows us to de ne, through an explicit trivialization, a smooth family of boundary conditions of generalized Atiyah-Patodi-Singer type. The calculus of b-pseudodi erential operators is then employed to establish the family index formula. A relative index formula, describing the e ect of changing the choice of the trivialization, is also given. In case the boundary family is invertible the form of the index theorem obtained by B i s m ut and Cheeger is recovered.

The Mathematical Work of V. K. Patodi

“…Thus, the η-form of Bismut-Cheeger generalizes the η-invariant of Atiyah-Patodi-Singer to the case of families. The natural question of whether there is a families generalization of the Atiyah-Patodi-Singer index theorem 3.1 was answered positively in [46,47] and [96]. On the other hand, the concept of spectral flow was generalized to the families index theory in [68].…”

“…The ideas and methods in [44][45][46][47]96], and [68] have been further generalized to the framework of noncommutative index theory. We refer to [89,90] for an overview.…”

An index theorem in differentialK–theory

“…We want to make the D V -term in the integrand invertible for large s , but we want to keep the small-s asymptotics of the unperturbed Bismut superconnection. As in [33, Section 5], we use the trick of "time-varying Á-forms", which originated in [37].…”

Bott periodicity for fibred cusp operators

“…Spectral sections were originally introduced in [17] to describe the boundary conditions for families of Dirac operators on manifolds with boundary. We intend to use spectral sections for a very different purpose, and instead of dealing with families of Dirac operators, we will consider families of Laplacians.…”