##
**An extension of a theorem by Cheeger and Müller. With an appendix by François Laudenbach.**
*(English)*
Zbl 0781.58039

Astérisque. 205. Paris: Société Mathématique de France, 235 p. (1992).

In what follows \(M\) is a compact manifold of dimension \(n\), \(F\) is a flat vector bundle on \(M\), \(H^*(M,F) = \bigoplus^ n_{i=0}H^ i(M,F)\) is the cohomology of the sheaf of locally flat sections of \(F\),
\[
\text{det }H^*(M,F) = \bigotimes^ n_{i=0}(\text{det }H^ i(M,F))^{(-1)^ i}
\]
is the real line defined to be the cohomology of \(F\) and \(g^ F\) and \(g^{TM}\) are smooth metrics of \(F\) and \(TM\) respectively. J. Cheeger [Ann. Math., II. Ser. 109, 259-322 (1979; Zbl 0412.58026)] proved a conjecture of D. B. Ray and I. M. Singer [Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)] suggesting that when the metric is flat, the Reidemeister and Ray-Singer metrics coincide. The result was proved also by W. Müller [Adv. Math. 28, 233-305 (1978; Zbl 0395.57011)] who slightly extended the result to the case when only the metric induced by \(g^ F\) on \(\text{det }F\) is flat [Analytic torsion and R-torsion for unimodular representations, Preprint MPI 91-50 (1991)]. The present work extends these results to the case when the metric on \(\text{det }F\) is not necessarily flat. The essential features of the work have already been announced in brief [the authors, C. R. Acad. Sci., Paris, Sér. I 313, No. 11, 775-782 (1991; Zbl 0743.57014)]. The present monograph gives all the details and contains a useful appendix on Thom-Smale complexes by F. Laudenbach.

In order to state the main result of the monograph we need the following additional notations: \(\|\;\|^{\text{RS}}_{\text{det }H\bullet(M,F)}\) and \(\|\;\|^{\text{R}}_{\text{det }H\bullet(M,F)}\) denote respectively the Ray-Singer and Reidemeister metrics, \(\nabla^{TM}\) denotes the Levi-Civita connection on \(TM\) corresponding to the metric induced on \(\text{det }F\) by \(g^ F\), \(\theta(F,g^ F)\) is the 1-form on \(M\) given by \[ \theta(F,g^ F) = \text{Tr}[\omega(F,g^ F)]\quad\text{where}\quad\omega(F,g^ F) = (g^ F)^{-1}\nabla^ Fg^ F \] and \(\nabla^ F\) is the connection corresponding to \(g^ F\). Further \(\psi(TM,\nabla^{TM})\) is the \(n-1\) current on \(TM\) that was constructed by V. Mathai and D. Quillen [Topology 25, 85-110 (1986; Zbl 0592.55015)], \(\|\;\|^{{\mathcal M},X}_{\text{det }H\bullet(M,F)}\) is the Milnor metric corresponding to the metric \(g^{F_ x}\) on \(F_ x\)’s (\(x\in B\)), where \(B\) is the set of zeros of \(X\) – a gradient vector field verifying the Smale transversality conditions. The main achievement of the work is the proof of the following identity: \[ \text{Log}\left({\|\;\|^{\text{RS}}_{\text{det }H\bullet(M,F)}\over \| \;\|^{{\mathcal M},X}_{\text{det }H \bullet (M,F)}}\right)^ 2 = - \int_ M\theta(F,g^ F)X^*\psi(TM,\nabla^{TM}). \] The first anomaly formulae for the Ray-Singer metrics are proved as essential tools for the proof of the main result. Several ramifications of the main and subsidiary results are considered.

In the appendix F. Laudenbach makes an attempt to unravel the essential simplicity underlying the hidden generic structure of the gradient field of a Morse function. In this context he develops a number of interesting properties of the Thom-Smale complex including the bifurcation of the complex in a 1-parameter family and some application of de Rham currents. The main fact proved in the appendix is that the pair \((f,X)\) of a function and a gradient vector field (with certain conditions) produces an embedding \(I_*\) of the Thom-Smale complex into the complex of de Rham currents, because the unstable manifolds of critical points are currents.

In order to state the main result of the monograph we need the following additional notations: \(\|\;\|^{\text{RS}}_{\text{det }H\bullet(M,F)}\) and \(\|\;\|^{\text{R}}_{\text{det }H\bullet(M,F)}\) denote respectively the Ray-Singer and Reidemeister metrics, \(\nabla^{TM}\) denotes the Levi-Civita connection on \(TM\) corresponding to the metric induced on \(\text{det }F\) by \(g^ F\), \(\theta(F,g^ F)\) is the 1-form on \(M\) given by \[ \theta(F,g^ F) = \text{Tr}[\omega(F,g^ F)]\quad\text{where}\quad\omega(F,g^ F) = (g^ F)^{-1}\nabla^ Fg^ F \] and \(\nabla^ F\) is the connection corresponding to \(g^ F\). Further \(\psi(TM,\nabla^{TM})\) is the \(n-1\) current on \(TM\) that was constructed by V. Mathai and D. Quillen [Topology 25, 85-110 (1986; Zbl 0592.55015)], \(\|\;\|^{{\mathcal M},X}_{\text{det }H\bullet(M,F)}\) is the Milnor metric corresponding to the metric \(g^{F_ x}\) on \(F_ x\)’s (\(x\in B\)), where \(B\) is the set of zeros of \(X\) – a gradient vector field verifying the Smale transversality conditions. The main achievement of the work is the proof of the following identity: \[ \text{Log}\left({\|\;\|^{\text{RS}}_{\text{det }H\bullet(M,F)}\over \| \;\|^{{\mathcal M},X}_{\text{det }H \bullet (M,F)}}\right)^ 2 = - \int_ M\theta(F,g^ F)X^*\psi(TM,\nabla^{TM}). \] The first anomaly formulae for the Ray-Singer metrics are proved as essential tools for the proof of the main result. Several ramifications of the main and subsidiary results are considered.

In the appendix F. Laudenbach makes an attempt to unravel the essential simplicity underlying the hidden generic structure of the gradient field of a Morse function. In this context he develops a number of interesting properties of the Thom-Smale complex including the bifurcation of the complex in a 1-parameter family and some application of de Rham currents. The main fact proved in the appendix is that the pair \((f,X)\) of a function and a gradient vector field (with certain conditions) produces an embedding \(I_*\) of the Thom-Smale complex into the complex of de Rham currents, because the unstable manifolds of critical points are currents.

Reviewer: C.S.Sharma (London)

### 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 |

53D50 | Geometric quantization |

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

57R99 | Differential topology |