We heard and learned from that Not all manifolds can be triangulated: In which dimensions?. "All orientable 5-dimensional manifolds are triangulable. In dimensions at least 6, though, you can use their construction to produce non-triangulable orientable manifolds."

What are some examples of non-triangulable manifolds which are orientable and non-orientable?

• Oreintable of non-triangulable manifolds, criteria and examples?

• Non-Oreintable of non-triangulable manifolds, criteria and examples?

• 4-dimensional E$_8$-manifold is non-triangulable. But it is a spin manifold. Is E$_8$-manifold triangulable or not? Why and how to prove this?