Aspherical manifolds that cannot be triangulated Joint work with Michael W. Davis, Jean-François Lafont
Kirby and Siebenmann showed that there are manifolds that do not admit PL structures , and yet the possibility remained that all manifolds could be triangulated, meaning that for every manifold , there is a simplicial complex so that the geometric realization of is homeomorphic to , but of course the simplicial complex is not a PL triangulation, meaning the links are not spheres.
Freedman showed that there are 4-manifolds that cannot be triangulated . Davis and Januszkiewicz applied a hyperbolization procedure to Freedman’s 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated . What about higher dimensions?
In the late 1970s Galewski and Stern  and independently, Matumoto, showed that non-triangulable manifolds exist in all dimensions if and only if homology -spheres with certain properties do not exist. In , Manolescu showed that there were no such homology -spheres, and hence non-triangulable manifolds exist in every dimension .
Applying a hyperbolization technique to the Galewski-Stern manifolds shows the following.
Theorem (Davis–Lafont–Fowler). Let . There exists a closed aspherical -manifold which cannot be triangulated.
However, the following question remains open: Do there exist closed aspherical -manifolds that cannot be triangulated?
 R.C. Kirby, L.C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969) 742–749. https://doi.org/10.1090/S0002-9904-1969-12271-8.
 M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geometry. 17 (1982) 357–453. http://projecteuclid.org/euclid.jdg/1214437136.
 M.W. Davis, T. Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347–388. http://projecteuclid.org/euclid.jdg/1214447212.
 D.E. Galewski, R.J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. Of Math. (2). 111 (1980) 1–34. https://doi.org/10.2307/1971215.
 C. Manolescu, The Conley index, gauge theory, and triangulations, J. Fixed Point Theory Appl. 13 (2013) 431–457. https://doi.org/10.1007/s11784-013-0134-3.