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 [1], and yet the possibility remained that all manifolds could be triangulated, meaning that for every manifold M , there is a simplicial complex K so that the geometric realization of K is homeomorphic to M , but of course the simplicial complex K is not a PL triangulation, meaning the links are not spheres.

Freedman showed that there are 4-manifolds that cannot be triangulated [2]. Davis and Januszkiewicz applied a hyperbolization procedure to Freedman’s 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated [3]. What about higher dimensions?

In the late 1970s Galewski and Stern [4] 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 [5], Manolescu showed that there were no such homology 3 -spheres, and hence non-triangulable manifolds exist in every dimension >4 .

Applying a hyperbolization technique to the Galewski-Stern manifolds shows the following.

Theorem (Davis–Lafont–Fowler). Let n > 5 . There exists a closed aspherical n -manifold which cannot be triangulated.

However, the following question remains open: Do there exist closed aspherical 5 -manifolds that cannot be triangulated?

[1] R.C. Kirby, L.C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969) 742–749.

[2] M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geometry. 17 (1982) 357–453.

[3] M.W. Davis, T. Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347–388.

[4] D.E. Galewski, R.J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. Of Math. (2). 111 (1980) 1–34.

[5] C. Manolescu, The Conley index, gauge theory, and triangulations, J. Fixed Point Theory Appl. 13 (2013) 431–457.

Download from the ArXiv