Pontrjagin classes and higher torsion of sphere bundles. (English) Zbl 1231.57025

Igusa, Kiyoshi

Penner, Robert (ed.) et al., Groups of diffeomorphisms in honor of Shigeyuki Morita on the occasion of his 60th birthday. Based on the international symposium on groups and diffeomorphisms 2006, Tokyo, Japan, September 11–15, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-48-8/hbk). Advanced Studies in Pure Mathematics 52, 21-29 (2008).

The author studies the relation between Pontrjagin classes and higher torsion invariants of a sphere bundle.
The concept of a sphere bundle is more or less equivalent to that of a Euclidean bundle, where the fibers are \({\mathbb R}^n\) but the structure group is \(\text{Homeo}({\mathbb R}^n,0)\). In contrast, the structure group for a vector bundle is \(GL(n)\) or even \(O(n)\). When the sphere bundle comes from a vector bundle, we call it linear.
A classical result defines rational Pontrjagin classes for an oriented Euclidean bundle, and hence for an oriented topological sphere bundle.
When the dimension of the fiber sphere is even, the higher Franz-Reidemeister torsion invariants are defined as transfers of multiples of certain Pontrjagin classes (the generalized Miller-Morita-Mumford classes), and hence are proportional to the corresponding topological Pontrjagin classes.
When the dimension of the fiber sphere is odd, but when the sphere bundle is linear, this proportionality still holds. However, Hatcher has constructed examples for a general sphere bundle where the differences between the higher torsion invariants and the multiple of topological Pontrjagin classes are not zero. The author defines this difference to be the exotic torsion. Therefore the exotic torsion measures the extent to which the odd-dimensional sphere bundle is not linear.
He then studies the exotic torsion in the framework of higher relative torsion invariants. In a joint work of the author with Goette whose details appear elsewhere, the exotic torsion is shown to be the same as the higher relative torsion for the sphere bundle \(E\) and a linear sphere bundle \(E_0\to E\) which is tangentially fiber homeomorphic to \(E\), and also proportional to the relative Dwyer-Weiss-Williams torsion defined in terms of their smoothing theory.
For the entire collection see [Zbl 1154.53004].

Reviewer: Zhaohu Nie (Altoona)

MSC:

57Q10	Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R20	Characteristic classes and numbers in differential topology

Keywords:

higher Franz-Reidemeister torsion; sphere bundles; topological Pontrjagin classes; exotic torsion; higher relative torsion