Two-Parameter Novikov-Shubin Invariants for Fibre Bundles

TL;DR

This paper introduces a two-parameter spectral density function and Novikov-Shubin invariants for fibre bundles, enabling a detailed analysis of how the spectra of fibres and bases influence the total space's spectrum.

Contribution

It develops a novel two-parameter generalization of spectral invariants for fibre bundles, preserving key invariance properties and providing explicit computations for specific groups.

Findings

The two-parameter spectral density function retains invariance properties.

Explicit computation for the three-dimensional Heisenberg group.

Enhanced understanding of spectral contributions from fibres and bases.

Abstract

In this paper we construct a two-parameter version of spectral density functions and Novikov-Shubin invariants on fibre bundles. The aim of this approach is to gain a better understanding of how the near-zero spectrum of the Hodge Laplace operators on the fibre and the base of a fibre bundle contribute separately to the near-zero spectrum of the Laplace operators of the total space. We show that this two-parameter generalisation of the classical spectral density function still satisfies several invariance properties. As an example, we compute it explicitly for the three-dimensional Heisenberg group.