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

平面矢量束的Bismut-Lott指数定理将光滑纤维束与紧致纤维束的总空间上的平面矢量束的Kamber-Tondeur类与基上的平面矢量束的Kamber-Tondeur类联系起来。由Dwyer， Weiss和Williams提出，这个指标定理仅仅对目前形式的拓扑纤维束失效。我们想把Bismut-Lott定理推广到更大的光滑映射类，从而得到奇异纤维束的上同调不变量。与BismutLott指标定理相关的是光滑纤维束的二次不变量，BismutLott高解析扭转。我们想把这个不变量与Igusa和Klein的拓扑定义的更高的Franz-Reidemeister扭转联系起来。作为一个应用程序，我们希望在许多情况下计算这两个不变量，并且我们希望使用它们来检测成对拓扑同构但不是差分同构的纤维束族。Heitsch和Lazarov将高解析扭转推广到叶形。我们想用叶片莫尔斯理论来计算这个扭转。作为一个应用，我们想要展示叶的家族是两两拓扑，但不是差分，同构。

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.