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 [MR242166], 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 [MR679066]. Davis and Januszkiewicz applied a hyperbolization procedure to Freedman's 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated [MR1131435]. What about higher dimensions?

In the late 1970s Galewski and Stern [MR558395] 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 [MR3122335], 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?

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Davis, M. W., Fowler, J., & Lafont, J.-F. (2014). Aspherical manifolds that cannot be triangulated. Algebr. Geom. Topol., 14(2), 795–803.