Higher complex torsion and the framing principle.

*(English)*Zbl 1083.57030
Mem. Am. Math. Soc. 835, 94 p. (2005).

The work makes a noticeable addition to the literature on the theory of higher Franz-Reidemeister (FR) torsion, it provides a valuable survey and makes a number of important new contributions. One of the new results is the Framing Principle which is used to compute the higher FR torsion. The original version of this Principle developed in the author’s book [Higher Franz-Reidemeister torsion. AMS/IP Studies in Advanced Mathematics. 31. Providence (2002; Zbl 1016.19001)] has a restriction on the birth-death singularities of oriented generalized Morse functions. In the paper under review this condition is removed and the Framing Principle is proven for all families of oriented generalized Morse functions. Another main result of the paper is a generalization of higher FR torsion for bundles with almost complex fibres.

The complex torsion of a bundle \(E\to B\) with almost complex fibre \(M\) is a sequence of cohomology classes \(\tau^{\mathbb{C}}_k(E, \zeta)_m\) defined using the equivariant higher FR-torsion of the vertical tangent lens space bundle \(S^{2n-1}(E)/{\mathbb Z}_m\). It is related to the generalized Miller-Morita-Mumford classes \[ T_k(E)= \text{tr}^E_B(k! ch_k(T^vE))\in H^{2k}(B;{\mathbb Z}) \] by the following formula \[ \tau^\mathbb{C}_k(E, \zeta)_m = \frac{1}{2} m^kL_{k+1}(\zeta) \frac{1}{k!}T_k(E) \] where \[ L_{k+1}(\zeta) = {\mathcal R}\left(\frac{1}{i^k}\sum_{n=1}^\infty \frac{\zeta^n}{n^{k+1}}\right) \] is the polylogarithm function. The main theorems of the paper are examples of the transfer theorem for higher FR-torsion. This theorem says (roughly) that the higher FR-torsion commutes with transfer.

The complex torsion of a bundle \(E\to B\) with almost complex fibre \(M\) is a sequence of cohomology classes \(\tau^{\mathbb{C}}_k(E, \zeta)_m\) defined using the equivariant higher FR-torsion of the vertical tangent lens space bundle \(S^{2n-1}(E)/{\mathbb Z}_m\). It is related to the generalized Miller-Morita-Mumford classes \[ T_k(E)= \text{tr}^E_B(k! ch_k(T^vE))\in H^{2k}(B;{\mathbb Z}) \] by the following formula \[ \tau^\mathbb{C}_k(E, \zeta)_m = \frac{1}{2} m^kL_{k+1}(\zeta) \frac{1}{k!}T_k(E) \] where \[ L_{k+1}(\zeta) = {\mathcal R}\left(\frac{1}{i^k}\sum_{n=1}^\infty \frac{\zeta^n}{n^{k+1}}\right) \] is the polylogarithm function. The main theorems of the paper are examples of the transfer theorem for higher FR-torsion. This theorem says (roughly) that the higher FR-torsion commutes with transfer.

Reviewer: Michael Farber (Zürich)

##### MSC:

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

57R45 | Singularities of differentiable mappings in differential topology |

57R50 | Differential topological aspects of diffeomorphisms |

19J10 | Whitehead (and related) torsion |

58J52 | Determinants and determinant bundles, analytic torsion |