CAT(0) metrics on contractible manifolds

TL;DR

This paper explores the relationship between CAT(0) metrics and the topology of high-dimensional manifolds, showing that certain geometric conditions imply topological collapsibility and vice versa.

Contribution

It establishes new connections between CAT(0) polyhedral metrics and topological properties like collapsibility and fundamental group stability in high-dimensional manifolds.

Findings

Manifolds with CAT(0) metrics are pseudo-collarable and have stable fundamental groups.

Such manifolds are topologically collapsible when dimension is at least 6.

Finite collapsible polyhedra are PL homeomorphic to CAT(0) cubical complexes.

Abstract

We prove that an open manifold $M$ of dimension at least $5$ which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann obstruction $\tau_{\infty}(M)$. Moreover, this implies that $M$ is topologically collapsible, when $n\geq 6$. Conversely, any finite dimensional collapsible polyhedron is PL homeomorphic to a CAT(0) cubical complex.