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

平坦向量丛的Bismut-Lott指标定理将具有紧纤维的光滑纤维丛的全空间上的平坦向量丛的Kamber-Tondeur类与作为基底上的向量丛的纤维上同调的Kamber-Tondeur类联系起来。由Dwyer，韦斯和威廉姆斯，这个指标定理失败，只是拓扑纤维丛在其目前的形式。我们要推广的Bismut-Lott定理更大类的光滑映射，从而获得上同调不变量的奇异纤维丛。与Bismut-Lott指数定理相关的是光滑纤维丛的一个次不变量，即Bismut-Lott高阶解析挠率。我们想把这个不变量与Igusa和Klein的拓扑定义的更高的Franz-Reidemeister挠率联系起来。作为一个应用程序，我们要计算这两个不变量在许多情况下，我们要用它们来检测家庭的纤维束是成对的拓扑，但不是差分，同构。Heitsch和Lazarov将更高的解析挠率推广到叶理。我们想用逐叶的莫尔斯理论来计算这个挠率。作为一个应用程序，我们要展示家庭的叶理，是成对的拓扑，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.