manifolds without symplicial or cell structure

Question

In many situations in topology, (like the poincare duality) they put a distinction between the space being a manifold or just a cell or simplicial complex. I want to know why this is important, in other words are there manifolds without the latter structure?

Answer 1

A homeomorphism from a manifold to a simplicial complex is called a triangularization. The $E_8$ manifold (a four-dimensional manifold) does not have a triangularization. I don't know the history of this result but I suspect it's from the late 80s as the proof uses Casson's invariant.

More recently -- just over a year ago! -- Ciprian Manolescu proved that there are manifolds of any dimension greater than 4 which do not admit an triangularization. He uses Pin(2)-equivariant Seiberg-Witten Floer homology. Here is the paper.