Real embeddings and eta invariants. (English) Zbl 0795.57010

Let \(i:Y \to X\) be an embedding of real compact odd dimensional manifolds. The purpose of this paper is to establish a formula mod\((\mathbb{Z})\) expressing the difference of certain eta invariants over \(X\) as the sum of an eta invariant over \(Y\) and of integrals over \(X\) and \(Y\) of Chern-Simons currents.


57R20 Characteristic classes and numbers in differential topology
58J20 Index theory and related fixed-point theorems on manifolds
57R40 Embeddings in differential topology
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[1] [ABoP] Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the Index Theorem. Invent. Math.19, 279-330 (1973) · Zbl 0257.58008
[2] [AH] Atiyah, M.F., Hirzebruch, F.: Riemann-Roch theorem for differential manifolds. Bull. Am. Math. Soc.65, 276-281 (1959) · Zbl 0142.40901
[3] [APS] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc.77, 43-69 (1975) · Zbl 0297.58008
[4] [B1] Bismut, J.M.: Superconnection currents and complex immersions. Invent. Math. 59-113 (1990) · Zbl 0696.58006
[5] [B2] Bismut, J.M.: Eta invariants and complex immersions. Bull. Soc. Math. France118, 211-227 (1990) · Zbl 0721.58057
[6] [B3] Bismut, J.M.: Koszul complexes, harmonic oscillators and the Todd class. J.A.M.S.3, 159-256 (1990) · Zbl 0702.58071
[7] [BC] Bismut, J.M., Cheeger, J.: ?-invariants and their adiabatic limits. J.A.M.S.2, 33-70 (1989) · Zbl 0671.58037
[8] [BF] Bismut, J.-M., Freed, D.S.: The analysis of elliptic families II. Dirac operators, eta invariants and the holonomy theorem. Comm. Math. Phys.107, 103-163 (1986) · Zbl 0657.58038
[9] [BGS] Bismut, J.M., Gillet, H., Soul?, C.: Bott-Chern currents and complex immersions. Duke Math. J.60, 255-284 (1990) · Zbl 0697.58005
[10] [BL] Bismut, J.M., Lebeau, G.: Complex immersions and Quillen metrics. Publ. Math. Inst. Hautes Etud. Sci.91, 1-298 (1991) · Zbl 0784.32010
[11] [Ge] Getzler, E.: A short proof of the Atiyah-Singer Index Theorem. Topology25, 111-117 (1986) · Zbl 0607.58040
[12] [GS] Gillet, H., Soul?, C.: Analytic torsion and the arithmetic Todd genus. Topology30, 21-54 (1991) · Zbl 0787.14005
[13] [Q] Quillen, D.: Superconnections and the Chern character. Topology24, 89-95 (1985) · Zbl 0569.58030
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