Generalized Morse Functions, Excision and Higher Torsions
Martin Puchol, Junrong Yan
TL;DR
This work advances the higher-analytic versus higher-topological torsion comparison by removing the fiberwise Morse function restriction. It introduces fiberwise generalized Morse functions and a double suspension to stabilize birth–death phenomena, and constructs a relative torsion framework that subtracts trivial-bundle contributions. The authors fuse analytic techniques (Witten deformation, Agmon estimates) with combinatorial torsion via a higher Igusa–Klein framework, proving a differential-form level equality of relative torsion forms under a generalized Thom–Smale transversality condition. The development is split into three parts: topological preliminaries and constructions (Part I), analytic estimates on noncompact settings (Part II), and the consolidation of intermediate results toward the generalized higher CM/BZ theorem (Part III). The results offer a pathway to transfer known properties between torsion invariants and connect to broader conjectures such as the transfer index conjecture, with potential implications for rigidity phenomena in torsion theory.
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-Müller/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.