Is there a lattice model of E8 manifold? 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?
 A: I'm not certain I understand your question, but it sounds like you might be looking for a piecewise-linear (PL) structure on the E8 manifold.  Roughly, this means than the manifold can be divided into 0-cells, 1-cells, ... 4-cells, and the local combinatorics of how the cells meet is tame/non-pathological/standard.  If that's your question, then the answer is no: in dimension 4, a topological manifold has a PL structure if and only if it has a smooth structure.
A: It might be useful to look at a construction of the E_8 manifold. It is glued from 2 parts, one which is very simple and has a  smooth structure and contains the intersection form, and the other one which is very mysterious. The simple one starts from the E_8 form and glues 8 copies of the tangent disc bundle of S^2 in a simple way so that the resulting manifold with boundary has intersection form E_8.  The  boundary is the Poincare homology sphere P. The mysterious part comes from Freedman's theory which implies that there is a compact contractible topological manifold with boundary P. The E_8-manifold is the union of the result of plumbing and the contractible topological manifold glued along P The second homology of the plumbed part maps isomorphically to the second homology of the E_8-manifold and so you can see the intersection form on the smooth part. 
