On a Hitchin-Thorpe inequality for manifolds with foliated boundaries

TL;DR

This paper extends the Hitchin-Thorpe inequality to noncompact 4-manifolds with foliated boundaries, using signature and rho invariant techniques to establish new geometric constraints.

Contribution

It introduces a Hitchin-Thorpe inequality for manifolds with foliated boundaries, expanding previous results to a broader geometric setting.

Findings

Derived a new inequality relating signature and Euler characteristic for foliated boundary manifolds.

Provided explicit examples illustrating the applicability of the inequality.

Extended the G-signature formula and rho invariant methods to noncompact foliated manifolds.

Abstract

We prove a Hitchin-Thorpe inequality for noncompact 4-manifolds with foliated geometry at infinity by extending on previous work by Dai and Wei. After introducing the objects at hand, we recall some preliminary results regarding the $G$-signature formula and the rho invariant, which are used to obtain expressions for the signature and Euler characteristic in our geometric context. We then derive our main result, and present examples.