Hyperbolization and regular neighborhoods

TL;DR

This paper demonstrates how hyperbolization affects regular neighborhoods of PL submanifolds and constructs negatively curved manifolds with specific topological properties, revealing new insights into geometric topology.

Contribution

It establishes a connection between hyperbolization and regular neighborhoods, and constructs novel negatively curved manifolds with particular homotopy and smoothness properties.

Findings

Hyperbolization pulls back regular neighborhoods of PL submanifolds.

Constructs open negatively curved manifolds homotopy equivalent to non-smoothable manifolds.

Shows existence of negatively pinched manifolds with specific topological features.

Abstract

We show that the hyperbolization of polyhedra pulls back regular neighborhoods of PL submanifolds. Applying this to the Riemannian version of the hyperbolization due to Ontaneda gives open complete manifolds of pinched negative curvature that are homotopy equivalent to closed smooth manifolds but contain no smooth spines. We also find open complete negatively pinched manifolds that are homotopy equivalent to closed non-smoothable manifolds.