$L^2$-torsion of fibrations

TL;DR

This paper investigates the $L^2$-torsion of fibrations, extending existing formulas to cases with relaxed conditions, and provides new sum and product formulas within an extended abelian category framework.

Contribution

It introduces sum and product formulas for $L^2$-torsion in the extended abelian category, relaxing previous acyclicity and determinant class assumptions.

Findings

Proved sum formula for $L^2$-torsion in fibrations.

Established product formula for $L^2$-torsion.

Derived $L^2$-torsion formula for simple fibrations with zero Euler characteristic.

Abstract

The paper studies the $L^2$-torsion of fibrations, focusing on cases that relax acyclicity and the determinant class condition. We prove the sum formula and the product formula for $L^2$-torsion in the extended abelian category. The desired formula for $L^2$-torsion of a simple fibration is obtained under the assumption that the fibers have zero Euler characteristic.