[Paper Review] Higher analytic torsion of sphere bundles and continuous cohomology of $Diff(S^{2n-1})$

Ulrich Bunke|ArXiv.org|Feb 20, 1998

Homotopy and Cohomology in Algebraic Topology5 references5 citations

TL;DR

This paper constructs higher analytic torsion classes for smooth sphere bundles using Bismut-Lott's torsion forms, and applies them to define nontrivial continuous group cohomology classes for the diffeomorphism group of odd-dimensional spheres, specifically $\mathrm{Diff}(S^{2n-1})$. The key result is the explicit computation of these classes in the case of sphere bundles arising from complex vector bundles, revealing new invariants in continuous cohomology.

ABSTRACT

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the diffeomorphism group of odd dimensional spheres.

Motivation & Objective

• To define characteristic classes for smooth sphere bundles using higher analytic torsion forms.
• To explore the continuous cohomology of the diffeomorphism group $\mathrm{Diff}(S^{2n-1})$ via geometric invariants.
• To compute these characteristic classes explicitly in the case where the sphere bundle arises from a complex vector bundle.
• To establish the nontriviality of the resulting cohomology classes in continuous group cohomology.
• To connect analytic torsion invariants with topological invariants of diffeomorphism groups of odd-dimensional spheres.

Proposed method

• Utilizes the higher analytic torsion form introduced by Bismut and Lott to construct characteristic classes for smooth sphere bundles.
• Applies the torsion form to sphere bundles obtained as sphere bundles of complex vector bundles.
• Constructs a characteristic class in de Rham cohomology that lifts to continuous group cohomology of $\mathrm{Diff}(S^{2n-1})$.
• Employs the Chern-Weil theory framework to relate curvature and characteristic classes in the context of fiber bundles.
• Uses the structure of the frame bundle and connection data on the total space to compute the torsion form.
• Relies on the theory of continuous group cohomology to interpret the resulting classes as invariants of the diffeomorphism group.

Experimental results

Research questions

• RQ1 Can higher analytic torsion be used to define characteristic classes for smooth sphere bundles?
• RQ2 Do these classes yield nontrivial invariants in the continuous cohomology of $\mathrm{Diff}(S^{2n-1})$?
• RQ3 What is the explicit form of the higher analytic torsion class when the sphere bundle arises from a complex vector bundle?
• RQ4 How do these classes relate to known characteristic classes in differential geometry?
• RQ5 Are the resulting cohomology classes stable or detectable under natural geometric constructions?

Key findings

• The higher analytic torsion form yields a well-defined characteristic class in de Rham cohomology for smooth sphere bundles.
• This characteristic class lifts to a nontrivial continuous group cohomology class of $\mathrm{Diff}(S^{2n-1})$.
• For sphere bundles derived from complex vector bundles, the torsion class is explicitly computable and nonvanishing.
• The construction provides a geometric realization of nontrivial continuous cohomology classes in degree 2n−1 for $\mathrm{Diff}(S^{2n-1})$.
• The result establishes a link between analytic torsion and the continuous cohomology of diffeomorphism groups of odd-dimensional spheres.
• The method demonstrates that higher analytic torsion can detect topological invariants beyond classical characteristic classes.