The Eta Invariant and Families of Pseudodifferential Operators

For a compact manifold without boundary a suspended algebra of pseudodifferential operators is considered; it is an algebra of pseudodifferential operators on, and translation-invariant in, an additional real variable. It is shown that the eta invariant, as defined by Atiyah, Patodi and Singer for admissible Dirac operators, extends to a homomorphism from the ring of invertible elements of the suspended algebra to the additive real line. The deformation properties of this extended eta homomorphism are discussed and a related ‘divisor flow’ is shown to label the components of the set of invertible elements within each component of the elliptic set. Introduction The eta invariant of the spin Dirac operator, and of the signature operator, on an odd dimensional manifold was introduced by Atiyah, Patodi and Singer [1] as the boundary correction term for their index formula on an even-dimensional compact manifold with boundary. Their definition extends directly to all ‘admissible’ Dirac operators and was later shown to extend to all self-adjoint elliptic pseudodifferential operators on compact manifolds without boundary. In the Dirac setting there are various further extensions to non-compact manifolds (by Brüning and Seeley [6], by Müller [16], by Stern [19] and in [13]), to singular manifolds (by Cheeger [8]) to boundary problems (by Branson and Gilkey [5], by Douglas and Wojciechowski [9], by Lesch and Wojciechowski [11] and by Müller [17]), to families (by Bismut and Cheeger [3], [2] and in [14], [15] ) also to define ‘higher’ eta invariants (by Lott [12], by Getzler [10] and by Wu [20]). Here a somewhat different ‘pseudodifferential’ extension of the eta invariant is given. This is closely related to Singer’s comments in [18] on the formal analogy between the index function and the eta invariant (despite their obvious differences). Received January 31, 1995. Supported in part by NSF grant 9306389-DMS. Manuscript available from ftp-math-papers.mit.edu