Aspherical manifolds that cannot be triangulated

TL;DR

This paper demonstrates that for each dimension greater than 5, there exist closed aspherical manifolds that cannot be triangulated, extending previous results in higher-dimensional topology.

Contribution

It applies hyperbolization techniques to Galewski-Stern manifolds to prove non-triangulability in all dimensions greater than 5.

Findings

Existence of non-triangulable aspherical manifolds in dimensions > 5

Extension of non-triangulability results to new classes of manifolds

Open question remains in dimension 5

Abstract

Although Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern and independently Matumoto showed that non-triangulable manifolds exist in all dimensions > 4 if and only if homology 3-spheres with certain properties do not exist. In 2013 Manolescu showed that, indeed, there were no such homology 3-spheres and hence, non-triangulable manifolds exist in each dimension >4. It follows from work of Freedman in 1982 that there are 4-manifolds that cannot be triangulated. In 1991 Davis and Januszkiewicz applied a hyperbolization procedure to Freedman's 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated. In this paper we apply hyperbolization techniques to the Galewski-Stern manifolds to show that there exist closed aspherical n-manifolds that cannot be triangulated for each n> 5. The question remains open in dimension 5.