**Background**

I'm using physics terminology because I'm not sure what the right mathematical terminology is, perhaps a simplicial complex?

I'm interested, for various physics reasons, in four manifolds and specifically in their intersection forms. I'm especially interested in the E8 manifold, but if I understand the situation correctly, this manifold is not smooth. I have even read statements that it cannot be triangulated in a certain sense. Obviously I can read the various definitions, but I don't have any intuition for them and I suspect getting the intuition would take me way too far afield. For my physics purposes I would have really liked this manifold to have a nice differential form cohomology, so I'm trying to understand what I can use instead.

**Main question**

As an example of a simpler structure that I could use, I would be happy with a lattice model of the E8 manifold. By "lattice model" I mean something like the way a large square lattice with periodic boundary conditions is a model of a torus. There is a discrete notion of points, links, and plaquettes so that the various topological properties are correctly captured. For example, I have in essence non-contractible loops and so forth.

Does something like this exist for the E8 manifold i.e. a discrete structure with the right intersection form, or is this impossible?

nothomeomorphic to a simplicial complex. If you has some combinatorial structure which captured the topological properties, I'm sure you could arrive at a triangulation. This is certainly the case for the 'lattice model' for the torus you describe. But this is just intuition on my part. $\endgroup$