Families torsion and Morse functions.

*(English)*Zbl 1071.58025
Astérisque 275. Paris: Société Mathématique de France (ISBN 2-85629-109-0/pbk). ix, 293 p. (2001).

The R-torsion is a topological invariant introduced by Reidemeister-Franz for manifolds equipped with a unitarily flat vector bundle. Later, Ray-Singer defined the analytic torsion, which is an analytic version of the R-torsion, and proved that both of these torsions are equal for lens spaces [D. B. Ray and I. M. Singer, Adv. Math. 7, 145–210 (1971; Zbl 0239.58014)]. They also conjectured that this equality holds in general, which was later proven by J. Cheeger [Ann. Math. (2) 109, 259–322 (1979; Zbl 0412.58026)] and W. Müller [Adv. Math. 28, No. 3, 233–305 (1978; Zbl 0395.57011)]. Further generalizations of this conjecture were obtained by J. Lott and M. Rothenberg [J. Differ. Geom. 34, No. 2, 431–481 (1991; Zbl 0744.57021)] and W. Lück [J. Differ. Geom. 37, No. 2, 263–322 (1993; Zbl 0792.53025)] for isometric group actions, and by J.-M. Bismut and W. Zhang for non-unitarily flat bundles [Astérisque 205 (1992; Zbl 0781.58039); Geom. Funct. Anal. 4, No. 2, 136–212 (1994; Zbl 0830.58030)].

Then Bismut-Lott constructed the higher analytic torsion for \(S^1\) fiber bundles equipped with a complex Hermitian line bundle with a unitarily flat connection, whose holonomy along the fibers is a root of unity [J.-M. Bismut and J. Lott, J. Am. Math. Soc. 8, No. 2, 291–363 (1995; Zbl 0837.58028)]; the zero degree component of this higher analytic torsion is given by the usual analytic torsion. On the other hand, by using algebraic \(K\)-theory, the higher R-torsion was defined by K. Igusa [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 643–651 (1991; Zbl 0755.58016)] and J. R. Klein [Proceedings of workshops held June 24–28, 1991, in Göttingen, Germany, and August 6–10, 1991, in Seattle, WA (USA), Contemp. Math. 150, 195–204 (1993; Zbl 0790.19006)]. Both of these higher torsions coincide for \(S^1\) fiber bundles over \(S^2\), but a general comparison formula seems to be very difficult to obtain.

In this paper, the authors prove many properties of the higher analytic torsion of Bismut-Lott: it is extended to the equivariant setting, a proper normalization is given, rigidity formulas are proven, it is evaluated (modulo coboundaries) for families of manifolds with a fiberwise Morse function, and a formula is given for the case of unit sphere bundles. This generalizes the results of Cheeger, Müller, Lott-Rothenberg and Bismut-Zhang on the relation between analytic torsion and R-torsion, as well as computations by U. Bunke for sphere bundles [Am. J. Math. 122, 377–401 (2000; Zbl 0984.58021)].

Then Bismut-Lott constructed the higher analytic torsion for \(S^1\) fiber bundles equipped with a complex Hermitian line bundle with a unitarily flat connection, whose holonomy along the fibers is a root of unity [J.-M. Bismut and J. Lott, J. Am. Math. Soc. 8, No. 2, 291–363 (1995; Zbl 0837.58028)]; the zero degree component of this higher analytic torsion is given by the usual analytic torsion. On the other hand, by using algebraic \(K\)-theory, the higher R-torsion was defined by K. Igusa [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 643–651 (1991; Zbl 0755.58016)] and J. R. Klein [Proceedings of workshops held June 24–28, 1991, in Göttingen, Germany, and August 6–10, 1991, in Seattle, WA (USA), Contemp. Math. 150, 195–204 (1993; Zbl 0790.19006)]. Both of these higher torsions coincide for \(S^1\) fiber bundles over \(S^2\), but a general comparison formula seems to be very difficult to obtain.

In this paper, the authors prove many properties of the higher analytic torsion of Bismut-Lott: it is extended to the equivariant setting, a proper normalization is given, rigidity formulas are proven, it is evaluated (modulo coboundaries) for families of manifolds with a fiberwise Morse function, and a formula is given for the case of unit sphere bundles. This generalizes the results of Cheeger, Müller, Lott-Rothenberg and Bismut-Zhang on the relation between analytic torsion and R-torsion, as well as computations by U. Bunke for sphere bundles [Am. J. Math. 122, 377–401 (2000; Zbl 0984.58021)].

##### MSC:

58J52 | Determinants and determinant bundles, analytic torsion |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

37D15 | Morse-Smale systems |

57R20 | Characteristic classes and numbers in differential topology |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J22 | Exotic index theories on manifolds |

57R22 | Topology of vector bundles and fiber bundles |

57R91 | Equivariant algebraic topology of manifolds |

57Q10 | Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. |