An interesting example for spectral invariants

TL;DR

This paper investigates spectral invariants related to heat operators of Dirac operators on foliations, demonstrating the optimality of previous bounds on Novikov-Shubin invariants through explicit examples.

Contribution

It constructs examples showing that the previously improved bounds on Novikov-Shubin invariants are sharp and cannot be further reduced.

Findings

Established the optimality of bounds on Novikov-Shubin invariants for heat operator convergence.

Provided explicit examples demonstrating the limits of current spectral invariant estimates.

Confirmed that previous improvements on invariants are the best possible.

Abstract

In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then showed that the associated heat operator converges to the Chern character of the index bundle of the operator. In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.