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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.

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
Full Text: DOI
[1] Adams, J.F., Vector fields on spheres, Ann. of math., 75, 3, 603-632, (1962) · Zbl 0112.38102
[2] Atiyah, M.; Patodi, V.; Singer, I., Spectral asymmetry and Riemannian geometry III, Math. proc. Cambridge philos. soc., 79, 71-99, (1976) · Zbl 0325.58015
[3] Atiyah, M.; Patodi, V.; Singer, I., Spectral asymmetry and Riemannian geometry II, Math. proc. Cambridge philos. soc., 78, 405-432, (1976) · Zbl 0314.58016
[4] Dai, X.; Zhang, W., Higher spectral flow, J. funct. anal., 157, 2, 432-469, (1998) · Zbl 0932.37062
[5] Gilkey, P.B., The residue of the global eta function at the origin, Adv. in math., 40, 290-307, (1981) · Zbl 0469.58015
[6] Gilkey, P.B., The eta invariant for even dimensional pinc manifolds, Adv. in math., 58, 243-284, (1985) · Zbl 0602.58041
[7] P.B. Gilkey, The Eta Invariant of Even Order Operators, Lecture Notes in Mathematics, Vol. 1410, Springer, Berlin, 1989, pp. 202-211. · Zbl 0820.58054
[8] Gilkey, P.B., The eta invariant of pin manifolds with cyclic fundamental groups, Periodica math. hungarica, 36, 139-170, (1998) · Zbl 0965.58024
[9] Karoubi, M., Algebres de Clifford et K-theorie, Ann. sci. ecole norm. sup., IV, 1, 161-270, (1968) · Zbl 0194.24101
[10] M. Karoubi, K-theory. An Introduction, Grundlehren der mathematischen Wissenschaften, Vol. 226, Springer, Berlin, 1978.
[11] Melrose, R., The Atiyah-patodi-Singer index theorem, research notes in mathematics, (1993), A.K. Peters Boston
[12] Moore, G.; Witten, E., Self-duality, Ramond-Ramond fields and K-theory, Jhep, 0005, 32, (2000) · Zbl 0990.81626
[13] Savin, A.; Schulze, B.-W.; Sternin, B., Elliptic operators in subspaces and eta invariant, K-theory, 26, 3, xx-xx, (2002), preliminary version at http://xxx.lanl.gov/abs/math/9907047
[14] Savin, A.Yu.; Sternin, B.Yu., Elliptic operators in even subspaces, Mat. sb., 190, 8, 125-160, (1999), (English transl.: Sb. Math. 190(8) (1999) 1195-1228) http://xxx.lanl.gov/abs/math/9907027 · Zbl 0963.58008
[15] Savin, A.Yu.; Sternin, B.Yu., Elliptic operators in odd subspaces, Mat. sb., 191, 8, 89-112, (2000), (English transl.: Sb. Math. 191(8) (2000)); http://xxx.lanl.gov/abs/math/9907039 · Zbl 0981.58018
[16] Savin, A.; Sternin, B., The eta-invariant and parity conditions, Dokl. RAN, 63, 2, 189-193, (2001) · Zbl 1044.58027
[17] Savin, A.; Sternin, B., The eta-invariant and Pontryagin duality in K-theory, Math. notes, 71, 2, 253-272, (2002), preliminary version is available at http://xxx.lanl.gov/abs/math/0006046 · Zbl 1030.58017
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