Asymptotic spectral flow for Dirac operators of disjoint Dehn twists

TL;DR

This paper investigates the asymptotic behavior of spectral flow for Dirac operators on 3-manifolds with contact forms, focusing on cases where the contact form's monodromy involves disjoint Dehn twists, revealing an order r growth in the spectral flow's next term.

Contribution

It provides a detailed analysis of spectral flow asymptotics for Dirac operators associated with Thurston-Winkelnkemper contact forms involving disjoint Dehn twists, identifying the order of the next term.

Findings

Spectral flow exhibits order r growth in the next term for the specified contact forms.

The analysis applies to Dirac operators on 3-manifolds with contact structures involving disjoint Dehn twists.

The results deepen understanding of spectral flow behavior in contact geometry and low-dimensional topology.

Abstract

Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is of order r.