# L^2 rho form for normal coverings of fibre bundles

## TL;DR

This paper introduces secondary invariants called L^2-eta and -rho forms for families of Dirac operators on normal coverings of fibre bundles, linking geometric analysis with scalar curvature metrics.

## Contribution

It defines new secondary invariants for Dirac operators on fiber bundle coverings, extending the analytical framework to non-compact settings with spectral assumptions.

## Key findings

- Defined L^2- eta and -rho forms for covering families.
- Established relations between L^2- rho classes and positive scalar curvature metrics.
- Analyzed large time asymptotics under spectral gap conditions.

## Abstract

We define the secondary invariants L^2- eta and -rho forms for families of generalized Dirac operators on normal coverings of fibre bundles. On the covering family we assume transversally smooth spectral projections, and Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time asymptotic for general operators. In the particular case of a bundle of spin manifolds, we study the L^2- rho class in relation to the space of positive scalar curvature vertical metrics.