An index theorem in differential K–theory

Freed, Daniel S¹ ; Lott, John²

¹ Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257, USA
² Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA

Geometry & topology, Tome 14 (2010) no. 2, pp. 903-966.

Résumé

Let $\pi :X\to B$ be a proper submersion with a Riemannian structure. Given a differential $K$–theory class on $X$, we define its analytic and topological indices as differential $K$–theory classes on $B$. We prove that the two indices are the same.

Détail

DOI : 10.2140/gt.2010.14.903

Keywords: index theory, Dirac operator, differential $K$–theory

Bibliographie

[1] M F Atiyah, F Hirzebruch, Riemann–Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959) 276

[2] M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, from: "Proc. Sympos. Pure Math., Vol. III", Amer. Math. Soc. (1961) 7

[3] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43

[4] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71

[5] M F Atiyah, G B Segal, Equivariant $K$–theory and completion, J. Differential Geometry 3 (1969) 1

[6] M F Atiyah, I M Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963) 422

[7] M F Atiyah, I M Singer, The index of elliptic operators. IV, Ann. of Math. $(2)$ 93 (1971) 119

[8] P Baum, R G Douglas, $K$ homology and index theory, from: "Operator algebras and applications, Part I (Kingston, Ont., 1980)" (editor R V Kadison), Proc. Sympos. Pure Math. 38, Amer. Math. Soc. (1982) 117

[9] N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer (2004)

[10] J M Bismut, The Atiyah–Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1985) 91

[11] J M Bismut, J Cheeger, $\eta$–invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989) 33

[12] J M Bismut, D S Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys. 107 (1986) 103

[13] J M Bismut, W Zhang, Real embeddings and eta invariants, Math. Ann. 295 (1993) 661

[14] U Bunke, M Kreck, T Schick, A geometric description of smooth cohomology

[15] U Bunke, T Schick, Smooth $K$–theory, to appear in Astérisque

[16] J Cheeger, J Simons, Differential characters and geometric invariants, from: "Geometry and topology (College Park, Md., 1983/84)" (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 50

[17] X Dai, Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4 (1991) 265

[18] G Faltings, Lectures on the arithmetic Riemann–Roch theorem, Annals of Math. Studies 127, Princeton Univ. Press (1992)

[19] D S Freed, On determinant line bundles, from: "Mathematical aspects of string theory (San Diego, Calif., 1986)" (editor S T Yau), Adv. Ser. Math. Phys. 1, World Sci. Publishing (1987) 189

[20] D S Freed, Dirac charge quantization and generalized differential cohomology, from: "Surveys in differential geometry" (editor S T Yau), Surv. Differ. Geom. VII, Int. Press (2000) 129

[21] D S Freed, Pions and generalized cohomology, J. Differential Geom. 80 (2008) 45

[22] D S Freed, M Hopkins, On Ramond–Ramond fields and $K$–theory, J. High Energy Phys. (2000) 14

[23] H Gillet, D Rössler, C Soulé, An arithmetic Riemann–Roch theorem in higher degrees, Ann. Inst. Fourier (Grenoble) 58 (2008) 2169

[24] H Gillet, C Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. II, Ann. of Math. $(2)$ 131 (1990) 205

[25] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[26] F Hirzebruch, Topological methods in algebraic geometry, Classics in Math., Springer (1995)

[27] M J Hopkins, M A Hovey, Spin cobordism determines real $K$–theory, Math. Z. 210 (1992) 181

[28] M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and $M$–theory, J. Differential Geom. 70 (2005) 329

[29] L Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Math., Springer (2003)

[30] M Karoubi, $K$–théorie multiplicative et homologie cyclique, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986) 507

[31] M Karoubi, Homologie cyclique et $K$–théorie, Astérisque 149, Soc. Math. France (1987) 147

[32] K Klonoff, An index theorem in differential $K$–theory, PhD thesis, University of Texas (2008)

[33] J Lott, $\mathbf{R}/\mathbf{Z}$ index theory, Comm. Anal. Geom. 2 (1994) 279

[34] J Lott, Secondary analytic indices, from: "Regulators in analysis, geometry and number theory" (editors A Reznikov, N Schappacher), Progr. Math. 171, Birkhäuser (2000) 231

[35] J Lott, Higher-degree analogs of the determinant line bundle, Comm. Math. Phys. 230 (2002) 41

[36] J P May, A concise course in algebraic topology, Chicago Lectures in Math., Univ. of Chicago Press (1999)

[37] R B Melrose, P Piazza, Families of Dirac operators, boundaries and the $b$–calculus, J. Differential Geom. 46 (1997) 99

[38] J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press (1974)

[39] A S Miščenko, A T Fomenko, The index of elliptic operators over $C^* $–algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 831, 967

[40] M Ortiz, Differential equivariant $K$–theory

[41] D Quillen, Determinants of Cauchy–Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985) 37, 96

[42] D Quillen, Superconnections and the Chern character, Topology 24 (1985) 89

[43] J Simons, D Sullivan, Structured vector bundles define differential $K$–theory

[44] C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press (1994)

[45] Z I Yosimura, Universal coefficient sequences for cohomology theories of $\mathrm{CW}$–spectra, Osaka J. Math. 12 (1975) 305