On the mean Euler characteristic of contact manifolds

TL;DR

This paper derives a formula for the mean Euler characteristic of contact manifolds using mean indices of Reeb orbits, exploring its behavior under surgeries and in Morse-Bott cases.

Contribution

It introduces a broad class of asymptotically finite contact manifolds and analyzes the mean Euler characteristic within this framework, including surgery effects.

Findings

Mean Euler characteristic expressed via mean indices of Reeb orbits

Class of asymptotically finite contact manifolds is closed under subcritical surgery

Derived formula for the Morse-Bott case

Abstract

We express the mean Euler characteristic of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is closed under subcritical surgery and examine the behavior of the mean Euler characteristic under such surgery. To this end, we revisit the notion of index-positivity for contact forms. We also obtain an expression for the mean Euler characteristic in the Morse-Bott case.