Mathematical Research Letters

Volume 2 (1995)

Number 5

The eta invariant and families of pseudodifferential operators

Pages: 541 – 561



Richard B. Melrose (Massachusetts Institute of Technology)


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.

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