Higher Torsion Invariants and Applications to Smooth Maps, Bundles and Foliations

点击翻译按钮获取中文摘要

The Bismut-Lott index theorem for flat vector bundles relates the Kamber-Tondeur classes of a flat vector bundle on the total space of a smooth fibre bundle with compact fibres to the Kamber-Tondeur classes of the fibre-wise cohomology as a vector bundle on the base. By Dwyer, Weiss and Williams, this index theorem fails for merely topological fibre bundles in its present form. We want to generalise the Bismut-Lott theorem to larger classes of smooth maps, thus obtaining cohomological invariants of singular fibre bundles. Associated to the BismutLott index theorem is a secondary invariant of smooth fibre bundles, the Bismut-Lott higher analytic torsion. We want to relate this invariant to the topologically defined higher Franz-Reidemeister torsion of Igusa and Klein. As an application, we want to compute both invariants in many cases, and we want to use them to detect families of fibre-bundles that are pairwise topologically, but not differentially, isomorphic. Heitsch and Lazarov generalised the higher analytic torsion to foliations. We want to use leaf-wise Morse theory to compute this torsion. As an application we want to exhibit families of foliations that are pairwise topologically, but not differentially, isomorphic.