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$ \eta$-invariants and their adiabatic limits

Authors: Jean-Michel Bismut and Jeff Cheeger
Journal: J. Amer. Math. Soc. 2 (1989), 33-70
MSC: Primary 58G10; Secondary 58C50, 58G12, 58G20
MathSciNet review: 966608
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  • [A] M. F. Atiyah, The logarithm of the Dedekind $ \eta $-function, Math. Ann. 278 (1987), 335-380. MR 909232 (89h:58177)
  • [ADS] M. F. Atiyah, H. Donnelly, and I. M. Singer, Eta invariants, signature defects of cusps, and values of $ L$-functions, Ann. of Math. (2) 118 (1983), 131-177. MR 707164 (86g:58134a)
  • [APS] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69. MR 0397797 (53:1655a)
  • [AS] M. F. Atiyah and I. M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 2597. MR 742394 (86g:58127)
  • [BaM] D. Barbasch and H. Moscovici, $ {L_2}$-index theorem and the Selbert trace formula, J. Funct. Anal. 53 (1983), 151-201. MR 722507 (85j:58137)
  • [B1] J.-M. Bismut, The Atiyah-Singer Index Theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), 91-151. MR 813584 (87g:58117)
  • [B2] -, Transgressed Chern forms for Dirac operators, J. Funct. Anal. 77 (1988), 32-50. MR 930389 (89j:58123)
  • [BC1] J. M. Bismut and J. Cheeger, Invariants êta et indice des familles pour des varietes a bord, C. R. Acad. Sci. Paris Sér I. 305 (1987), 127-130. MR 901625 (88g:58171)
  • [BC2] -, Families index for manifolds with boundary, superconnections and cones, J. Funct. Anal. (to appear).
  • [BF] J. M. Bismut and D. S. Freed, The analysis of elliptic families: Dirac operators, eta invariants, and the holonomy theorem of Witten, Comm. Math. Phys. 107 (1986), 103-163. MR 861886 (88h:58110b)
  • [BGS] J. M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles, II; Direct images and Bott-Chern forms, Comm. Math. Phys. 115 (1988), 79-126. MR 929147 (89g:58192b)
  • [C1] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. USA, Vol. 76, No. 5, pp. 2103-2106, May 1979. MR 530173 (80k:58098)
  • [C2] -, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657. MR 730920 (85d:58083)
  • [C3] -, $ \eta $-invariants, the adiabatic approximation and conical singularities, J. Differential Geom. 26 (1987), 175-221. MR 892036 (89c:58123)
  • [Chou] A. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvature, Trans. Amer. Math. Soc. 28 (1985), 1-40. MR 779050 (86i:58124)
  • [Co] A. Connes, Noncommutative differential geometry, Inst. Haute Études Sci. Publ. Math. 62 (1986) 41-144. MR 823176 (87i:58162)
  • [CoM] A. Connes and H. Moscovici, Transgression of the Chern character and cyclic cohomology, C. R. Acad. Sci. Paris Sér. I 303 (1986), 913-918. MR 873393 (88c:58006)
  • [D] X. Dai, Ph.D. thesis, SUNY at Stony Brook (in preparation).
  • [G] E. Getzler, A short proof of the Atiyah-Singer Index Theorem, Topology 25 (1986), 111-117. MR 836727 (87h:58207)
  • [Gi1] P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Math. Lecture Series., No. 11, Publish or Perish, Boston, 1984. MR 783634 (86j:58144)
  • [M1] W. Müller, Signature defects of cusps of Hilbert modular varieties and values of $ L$-series at $ s = 1$, J. Differential Geometry 20 (1984), 55-119. MR 772126 (87c:11048)
  • [M2] -, Manifolds with cusps of rank one: spectral theory and $ {L_2}$-index theorem, Lecture Notes in Math., no. 1244, Springer-Verlag, New York, 1987. MR 891654 (89g:58196)
  • [Q] D. Quillen, Superconnections and the Chern character, Topology 24 (1985), 89-95. MR 790678 (86m:58010)
  • [S] M. Stern, $ {L^2}$-index theorems on locally symmetric spaces (preprint).
  • [W] E. Witten, Global gravitational anomalies, Comm. Math. Phys. 100 (1985), 197-229. MR 804460 (87k:58282)

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