This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifert data.
As an application, we compute the Eells–Kuiper and t-invariants of certain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomorphism type of a new manifold of positive sectional curvature.
The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form on the determinant line of the cohomology. Both and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsion is defined it is equal to . As an application we obtain new results about the Burghelea-Haller torsion. In particular, we prove a weak version of the Burghelea-Haller conjecture relating...
Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.
In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As an application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms of its holonomy representation.
We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from to with p,q > 1. When 1 + n/p - n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n+1)/2 - n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou’s formula for the L² index. For 1 + n/p - n/q ≤ 0 the index formula contains an extra term related to the Calderón projector.