The signature of cusped hyperbolic 4-manifolds

TL;DR

This paper demonstrates that all integers can be realized as the signature of non-compact hyperbolic 4-manifolds with finite volume, using a combination of existing theorems and explicit constructions.

Contribution

It establishes that every integer is the signature of some hyperbolic 4-manifold and explores the topological properties of these manifolds.

Findings

Every integer is realizable as a hyperbolic 4-manifold signature

Constructed a hyperbolic 24-cell manifold with special properties

Partial results on the geography of hyperbolic 4-manifolds

Abstract

In this note we show that every integer is the signature of a non-compact, oriented, hyperbolic 4-manifold of finite volume, and give some partial results on the geography of such manifolds. The main ingredients are a theorem of Long and Reid, and the explicit construction of a hyperbolic 24-cell manifold with some special topological properties.