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. Then, is E$_8$-manifold oreintable or not? Why and how to prove this?

It looks that the 4-dimensional topological spin cobordism gives rise to an integer $\mathbb{Z}$ classification. While the E$_8$-manifold should be a manifold generator of the $\mathbb{Z}/2\mathbb{Z}$ classification instead. In contrast, the 4-dimensional topological pin$^+$ bordism gives rise to a $\mathbb{Z}/8\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ classification, and the 4-dimensional topological pin$^-$ bordism gives rise to a $\mathbb{Z}/2\mathbb{Z}$ classification. In this sense, it looks that the E$_8$-manifold should be a manifold generator of the $\mathbb{Z}/2\mathbb{Z}$ classification of both topological non-orientable pin$^+$ and pin$^-$ bordism, but not the orientable topological spin bordism. Then, should E$_8$-manifold be not orientable?