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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)].

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.