The eta invariant and parity conditions. (Q1424497)

On a compact manifold let \(A\) be an admissible self-adjoint elliptic pseudodifferential operator of positive order. (A pseudodifferential operator is admissible if the homogeneous components of its symbol behave, under multiplication of contangent vectors by \(-1\), in the same way as polynomial symbols.) \textit{P. B. Gilkey} in [Differential geometry, Proc. 3rd Int. Symp., Peniscola, Spain 1988, Lect. Notes Math. 1410, 202--211 (1989; Zbl 0820.58054)] observed that if the sum of the operator's order and the manifold's dimension is odd, then the fractional part of the operator's eta invariant is a homotopy invariant. Gilkey knew that this invariant is nontrivial for operators of odd order. The paper under review exhibits even-order operators for which the invariant is nontrivial. The paper also provides formulas for the invariant. One formula involves the \(K\)-theoretic push-forward of the symbol class of a related operator. This formula arises from the relationships among the fractional part of the operator's eta invariant, mod \(n\) spectral flow of a family constructed from the operator, and the index of an operator related to the family defining the spectral flow. Another formula expresses the invariant in terms of the \(K\)-theoretic linking pairing of a class represented by the original operator's symbol and a class constructed using the manifold's orientation bundle.