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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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