Bulletin of the American Mathematical Society

Author(s): Steven Rosenberg
Journal: Bull. Amer. Math. Soc. 34 (1997), 423-433.
MSC (1991): Primary 58G25; Secondary 58G10, 58G25, 58G26
Retrieve article in: PDF DVI PostScript

Abstract: In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Index Theorem has many applications and extensions to differential geometry, K-theory, mathematical physics, and other fields. This report focuses on advances in geometric aspects of index theory. For operators naturally associated to a Riemannian metric on a closed manifold, the topological side of the Index Theorem can often be expressed as the integral of local (i.e. pointwise) curvature expression. We will first discuss these local refinements in §1, which arise naturally in heat equation proofs of the Index Theorem. In §§2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlocal. Finally, in §4 we point out some recent connections among these nonlocal invariants and classical index theory.

1.: M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Inventiones Math. 19 (1973), 279-330. MR 58:31287
2.: M. F. Atiyah, H. Donnelly, and I. M. Singer, Eta invariants, signature defects of cusps and values of L-functions, Ann. Math. 118 (1983), 131-177. MR 86g:58134a
3.: M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. MR 53:1655a
4.: -, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71-99. MR 53:1655c
5.: M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433. MR 28:626
6.: -, The index of elliptic operators. III, Ann. Math. 87 (1968), 546-604. MR 38:5245
7.: -, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci., USA 81 (1984), 2597-2600. MR 86g:58127
8.: N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Grundlehren der mathematischen Wissenschaften, vol. 298, Springer-Verlag, Berlin, 1992. MR 94e:58130
9.: J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Inventiones Math. 83 (1986), 91-151. MR 87g:58117
10.: J.-M. Bismut and J. Cheeger, Families index for manifolds with boundary, superconnections, and cones. I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal. 89 (1990), 313-363. MR 91e:58180
11.: -, Families index for manifolds with boundary, superconnections, and cones. II. The Chern character, J. Funct. Anal. 90 (1990), 306-354. MR 91e:58181
12.: -, Transgressed Euler classes of SL(2n,Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. scient. Éc. Norm. Sup. 25 (1992), 335-391. MR 94e:57042
13.: J.-M. Bismut and D. Freed, The analysis of elliptic families I: Metrics and connections on determinant line bundles, Commun. Math. Phys. 106 (1986), 159-176. MR 88h:58110a
14.: -, The analysis of elliptic families II: Dirac operators, eta invariants, and the holonomy theorem of Witten, Commun. Math. Phys. 107 (1986), 103-163. MR 88h:58110b
15.: J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I-III, Commun. Math. Phys. 115 (1988), 49-78, 79-126, 301-351. MR 89g:58192a/b/c
16.: J.-M. Bismut and J. Lott, Flat vector bundles, direct images, and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), 291-363. MR 96g:58202
17.: J.-M. Bismut and W. Zhang, An extension of a theorem of Cheeger and Müller, Astérisque 205 (1992), 3-235. MR 93j:58138
18.: T. Branson and B. Ørsted, Conformal geometry and global invariants, Diff. Geom. Appl. 1 (1991), 279-308. MR 94k:58154
19.: D. Burghelea, L. Friedlander, and T. Kappeler, Torsions for manifolds with boundary and gluing formulas, preprint.
20.: A. Carey and V. Mathai, ${L}^2$ -acyclicity and ${L}^2$ -torsion invariants, Contemporary Mathematics 105 (1990), American Mathematical Society, Providence, RI, pps. 91-118. MR 91e:58187
21.: J. Cheeger, Analytic torsion and the heat equation, Ann. Math. 109 (1979), 259-322. MR 80j:58065a
22.: X. Dai and D. Freed, $\eta$ -invariants and determinant lines, J. Math. Phys. 35 (1994), 5155-5194. MR 96a:58204
23.: W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic K-theory Euler class, preprint, http:/www.math.uiuc.edu/K-theory/0086/(1995).
24.: E. Getzler, A short proof of the local Atiyah-Singer Index Theorem, Topology 25 (1986), 111-117. MR 87h:58207
25.: P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. in Math. 10 (1973), 344-381. MR 48:3081
26.: -, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Wilmington, DE, 1984. MR 86j:58144
27.: J. Heitsch and C. Lazarov, Riemann-Roch-Grothendieck and torsion for foliations, preprint.
28.: K. Igusa, Parametrized Morse theory and its applications, Proc. Int. Cong. Math., Kyoto 1990, Mathematical Society of Japan, Tokyo (1991), 643-651. MR 93c:57022
29.: F. Kamber and P. Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51-89. MR 48:12556
30.: J. Klein, Higher Franz-Reidemeister torsion: low-dimensional applications, Contemporary Mathematics 150 (1993), American Mathematical Society, Providence, RI, pps. 195-204. MR 94g:19004
31.: J. Lott, Diffeomorphisms, analytic torsion and noncommutative geometry, dg-ga/9607006.
32.: -, Heat kernels on covering spaces and topological invariants, J. Differential Geometry 35 (1992), 471-510. MR 93b:58140
33.: R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, Wellesley, MA, 1993. MR 96g:58180
34.: J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 33:4922
35.: W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233-305. MR 80j:58065b
36.: -, L ${}^2$ index theory, eta invariants and values of L-functions, Contemporary Mathematics 105 (1990), American Mathematical Society, Providence, RI, pps. 141-190. MR 91g:58274
37.: B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. MR 90d:58159
38.: R. Palais (ed.), Seminar on the Atiyah-Singer Index Theorem, Annals of Mathematics Study, vol. 57, Princeton University Press, Princeton, 1965. MR 33:6649
39.: T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geometry 25 (1987), 199-222. MR 89e:58118
40.: V. K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kähler manifolds, J. Differential Geometry 5 (1971), 251-283. MR 44:7502
41.: D. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funk. Anal. i Prilozhen 19 (1985), 37-41. MR 86g:32035
42.: -, Superconnections and the Chern character, Topology 24 (1985), 89-95. MR 86m:58010
43.: D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. MR 45:4447
44.: J. Roe, Elliptic Operators, Topology, and Asymptotic Methods, Pitman Research Notes in Mathematics, vol. 179, Longman Scientific and Technical, Burnt Mill, UK, 1988. MR 89j:58126
45.: -, Coarse cohomology and index theory on complete Riemannian manifolds, Memoirs of the Amer. Math. Soc., 497, 1993. MR 94a:58193
46.: S. Rosenberg, The determinant of a conformally covariant operator, J. London Math. Soc. 36 (1987), 553-568. MR 89h:58205
47.: -, The Laplacian on a Riemannian Manifold, Cambridge U. Press, Cambridge, UK, 1997.
48.: R. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math., vol. 10, Amer. Math. Soc., Providence, RI, 1967, pp. 288-307. MR 38:6220
49.: I. M. Singer, The $\eta$ -invariant and the index, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific Press, Singapore, 1987, pp. 239-258. CMP 20:04
50.: S. Vishik, Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary, Commun. Math. Phys. 167 (1995), 1-102. MR 96f:58184
51.: E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985), 197-229. MR 87k:58282
52.: -, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), 661-692. MR 84b:58111

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 58G25, 58G10, 58G25, 58G26

Steven Rosenberg
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email: sr@math.bu.edu

DOI: 10.1090/S0273-0979-97-00731-3
PII: S 0273-0979(97)00731-3
Additional Notes: Partially supported by the NSF
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS