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The eta invariant and parity conditions. (English) Zbl 1043.58012
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.) 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.

##### MSC:
 58J28 Eta-invariants, Chern-Simons invariants 19K56 Index theory 58J20 Index theory and related fixed-point theorems on manifolds 19L64 Geometric applications of topological $$K$$-theory
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