Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary

TL;DR

This paper proves a generalized Ray-Singer conjecture for manifolds with boundary, establishing the equality of combinatorial and analytic torsion norms under various boundary conditions without eigenvalue computations.

Contribution

It extends the Ray-Singer conjecture to manifolds with boundary, providing a new proof that avoids eigenvalue calculations and introduces new properties of analytic torsion.

Findings

The ratio between analytic and combinatorial torsion is explicitly computed for manifolds with boundary.

New properties of Ray-Singer analytic torsion are established.

The proof does not rely on eigenvalue asymptotics or explicit torsion formulas.

Abstract

This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjecture \cite{RS} claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal. For a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed. Some new general properties of the Ray-Singer analytic torsion are found. The proof does~not use any computation of eigenvalues and its asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds.