# zbMATH — the first resource for mathematics

Flat vector bundles, direct images and higher real analytic torsion. (English) Zbl 0837.58028
The purpose of the paper is to extend the Ray-Singer analytic torsion from an invariant of a smooth manifold to an invariant of a smooth parametrized family of manifolds. Let $Z\to M@>\pi>> B$ be a fiber bundle with closed fibers $$Z_b= \pi^{-1}(b)$$. To a flat complex vector bundle $$F$$ on $$M$$ one can associate certain characteristic classes $$c_k(F)\in H^k(M; \mathbb{R})$$ [see J. L. Dupont, Topology 15, 233-245 (1976; Zbl 0331.55012)]. Let $$e(TZ)\in H^{\dim(Z)}(M; \mathbb{R})$$ denote the Euler class of the vertical tangent bundle of the fibration. Let $$H^p(Z, F|_Z)$$ be the flat complex vector bundle on $$B$$ whose fiber over $$b\in B$$ is isomorphic to $$H^p(Z_b, F|_{Z_b})$$. The following smooth analog of the Riemann-Roch-Grothendieck theorem is shown:
Theorem. For any positive odd integer $$k$$, $\sum^{\dim(Z)}_{p= 0} (- 1)^p c_k(H^p(Z, F|_Z))= \int_Z e(TZ) c_k(F)\in H^k(B; \mathbb{R}).$ This follows from a more refined differential form version. On the level of differential forms the difference of the left- hand side and the right-hand side is shown to equal the exterior derivative of the so-called “higher analytic torsion form”. This is a certain differential form on $$B$$ of mixed even degree whose degree-zero part is the function which assigns to each $$b\in B$$ the Ray-Singer torsion of $$(Z_b, F|_{Z_b})$$.
The results of this paper have been announced by the authors in C. R. Acad. Sci., Paris, Sér. I 316, No. 5, 477-482 (1993; Zbl 0780.57023).
Reviewer: C.Bär (Freiburg)

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 57R57 Applications of global analysis to structures on manifolds 57R19 Algebraic topology on manifolds and differential topology
Full Text:
##### References:
  Jean-Michel Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), no. 1, 91 – 151. · Zbl 0592.58047 · doi:10.1007/BF01388755 · doi.org  Jean-Michel Bismut and Jeff Cheeger, \?-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), no. 1, 33 – 70. · Zbl 0671.58037  Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001  Alain Berthomieu and Jean-Michel Bismut, Quillen metrics and higher analytic torsion forms, J. Reine Angew. Math. 457 (1994), 85 – 184. · Zbl 0804.32017 · doi:10.1515/crll.1994.457.85 · doi.org  J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107 – 133. · Zbl 0354.55009  J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion, Comm. Math. Phys. 115 (1988), no. 1, 49 – 78. Jean-Michel Bismut, Henri Gillet, and Christophe Soulé, Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms, Comm. Math. Phys. 115 (1988), no. 1, 79 – 126. Jean-Michel Bismut, Henri Gillet, and Christophe Soulé, Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants, Comm. Math. Phys. 115 (1988), no. 2, 301 – 351. · Zbl 0651.32017  Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. · Zbl 0744.58001  Jean-Michel Bismut and Kai Köhler, Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom. 1 (1992), no. 4, 647 – 684. · Zbl 0784.32023  Jean-Michel Bismut and Gilles Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. 74 (1991), ii+298 pp. (1992). · Zbl 0784.32010  Jean-Michel Bismut and John Lott, Fibrés plats, images directes et formes de torsion analytique, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 5, 477 – 482 (French, with English and French summaries). · Zbl 0780.57023  Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235 – 272 (1975). · Zbl 0316.57026  Raoul Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71 – 112. · Zbl 0148.31906 · doi:10.1007/BF02391818 · doi.org  J. M. Bismut and W. Zhang, An extension of the Cheeger-Müller theorem, Astérisque, no. 205, Soc. Math. France, Paris, 1992.  Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259 – 322. , https://doi.org/10.2307/1971113 Werner Müller, Analytic torsion and \?-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233 – 305. · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0 · doi.org  Jeff Cheeger and James Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50 – 80. · Zbl 0621.57010 · doi:10.1007/BFb0075216 · doi.org  Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233 – 245. · Zbl 0331.55012 · doi:10.1016/0040-9383(76)90038-0 · doi.org  X. Dai and R. Melrose (to appear).  R. Forman (to appear).  W. Franz, Uber die Torsion einer überdeckrung, J. Reine Angew. Math. 173 (1935), 245-254. · JFM 61.1350.01  Kiyoshi Igusa, Parametrized Morse theory and its applications, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 643 – 651. · Zbl 0755.58016  John R. Klein, Higher Franz-Reidemeister torsion: low-dimensional applications, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 195 – 204. · Zbl 0790.19006 · doi:10.1090/conm/150/01291 · doi.org  Franz W. Kamber and Philippe Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51 – 89. · Zbl 0267.57012 · doi:10.1007/BF01189091 · doi.org  L. Lewin et al., Structural properties of polylogarithms, Amer. Math. Soc., Providence, RI, 1991. · Zbl 0745.33009  Jean-Louis Loday, Les matrices monomiales et le groupe de Whitehead \?\?$$_{2}$$, Algebraic \?-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 155 – 163. Lecture Notes in Math., Vol. 551 (French). · Zbl 0348.55007  J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358 – 426. · Zbl 0147.23104  Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259 – 322. , https://doi.org/10.2307/1971113 Werner Müller, Analytic torsion and \?-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233 – 305. · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0 · doi.org  Werner Müller, Analytic torsion and \?-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), no. 3, 721 – 753. · Zbl 0789.58071  Daniel Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89 – 95. · Zbl 0569.58030 · doi:10.1016/0040-9383(85)90047-3 · doi.org  -, Determinants of Cauchy-Riemann operators over a Riemann surface, Functional Anal. Appl. 14 (1985), 31-34. · Zbl 0603.32016  Daniel Quillen, Higher algebraic \?-theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 171 – 176. · Zbl 0359.18014  D. B. Ray and I. M. Singer, \?-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145 – 210. · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4 · doi.org  D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154 – 177. · Zbl 0267.32014 · doi:10.2307/1970909 · doi.org  K. Reidemeister, Homotopieringe und Linsenraüm, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109. · Zbl 0011.32404  R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288 – 307. · Zbl 0159.15504  Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979.  J. B. Wagoner, Diffeomorphisms, \?$$_{2}$$, and analytic torsion, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 23 – 33. · Zbl 0408.57015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.