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Description
Franz-Reidemeister and Ray-Singer torsions are two classical invariants of (unitarily) flat bundles over a manifold. They are defined topologically and analytically respectively. It was conjectured by Ray-Singer, and proved by the celebrated results of Cheeger, Müller and Bismut-Zhang, that we can compare these two torsions.
Consider now a family of manifolds with flat vector bundles. Then Bismut-Lott defined the so called analytic torsion form, an even differential form on the parameter manifold, which arises from a transgression in their Riemann-Roch-Grothendieck formula. Igusa and Klein constructed the topological counterpart of the analytic torsion form, known as the higher topological torsion. The relation between the analytic torsion form and the higher topological torsion is then a natural and important problem in the theory of higher torsion invariants, and a "higher" version of the Cheeger-Müller/Bismut-Zhang theorem is expected.
There are mainly two approaches to this problem: one due to Igusa, which uses axiomatization, and one due to Bismut-Goette, which uses Morse theory. After giving some background, we will see how Igusa's approach, along with results of various authors, can be used to compare the higher torsions associated with a trivial bundle. Building of this result, we can then use Bismut-Goette's approach to prove a relative comparison formula, which links a renormalized version of higher torsions. This renormalization consists in removing the torsion of a trivial bundle, and allows us to simplify some analytic difficulties arising when trying to extend the work of Bismut-Goette.