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$L^2$-torsion of fibrations

Chengzhang Sun

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.

$L^2$-torsion of fibrations

Abstract

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

Paper Structure

This paper contains 11 sections, 14 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Rudiments of L2-torsion
  4. Setup
  5. Determinant line and the torsion
  6. Combinatorial L2-torsion
  7. The sum formula and the product formula
  8. The sum formula
  9. The product formula
  10. L2-torsion of fibrations
  11. Open problems

Key Result

Lemma 2.1

Let $A,B$ be objects of $\mathcal{H}(A)$. The isomorphisms above are all orientation preserving.

Theorems & Definitions (28)

  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.3
  • Definition 2.4
  • Proposition 3.1
  • proof
  • ...and 18 more