Generalized Morse Functions, Excision and Higher Torsions

Abstract

Comparing invariants from both topological and geometric perspectives is a key focus in index theorem. This paper compares higher analytic and topological torsions and establishes a version of the higher Cheeger-M\"uller/Bismut-Zhang theorem. In fact, Bismut-Goette achieved this comparison assuming the existence of fiberwise Morse functions satisfying the fiberwise Thom-Smale transversality condition (TS condition). To fully generalize the theorem, we should remove this assumption. Notably, unlike fiberwise Morse functions, fiberwise generalized Morse functions (GMFs) always exist, we extend Bismut-Goette's setup by considering a fibration $ M \to S $ with a unitarily flat complex bundle $ F \to M $ and a fiberwise GMF $ f $, while retaining the TS condition. Compared to Bismut-Goette's work, handling birth-death points for a generalized Morse function poses a key difficulty. To deal with this, first, by the work of the author M.P., joint with Zhang and Zhu, we focus on a relative version of the theorem. Here, analytic and topological torsions are normalized by subtracting their corresponding torsions for trivial bundles. Next, using new techniques from by the author J.Y., we excise a small neighborhood around the locus where $f$ has birth-death points. This reduces the problem to Bismut-Goette's settings (or its version with boundaries) via a Witten-type deformation. However, new difficulties arise from very singular critical points during this deformation. To deal with these, we extend methods from Bismut-Lebeau, using Agmon estimates for noncompact manifolds developed by Dai and J.Y.