Complements of tori in $\#_{2k}S^2 \times S^2$ that admit a hyperbolic structure

TL;DR

This paper constructs specific hyperbolic link complements in certain 4-manifolds and establishes conditions for their existence, expanding understanding of hyperbolic structures in four-dimensional topology.

Contribution

It provides explicit examples of hyperbolic link complements in closed 4-manifolds and identifies necessary conditions for their presence.

Findings

Constructed hyperbolic link complements in $ abla_{2k}S^2 imes S^2$

Based on Ratcliffe and Tschantz's hyperbolic 4-manifolds

Established necessary conditions for hyperbolic link complements in simply connected 4-manifolds

Abstract

We construct examples of codimension two hyperbolic link complements in closed smooth 4-manifolds with homeomorphism type $\#_{2k}S^2 \times S^2$. All our examples are based on a construction of J. Ratcliffe and S. Tschantz, who constructed 1171 non-compact finite volume hyperbolic 4-manifolds of minimal volume. We then give necessary conditions for a closed smooth simply connected 4-manifold to contain a codimension two link complement that admits a hyperbolic structure.