We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least 6, though, you can use [Galewski and Stern's] construction to produce non-triangulable orientable manifolds."

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

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

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

• 4-dimensional $\mathrm E_8$-manifold is non-triangulable. But it is a spin manifold. Then, is $\mathrm E_8$-manifold orientable 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 $\mathrm E_8$-manifold should be a manifold generator of the $\mathbb{Z}/2\mathbb{Z}$ classification instead. In contrast, the 4-dimensional topological $\mathrm{pin}^+$ bordism gives rise to a $\mathbb{Z}/8\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ classification, and the 4-dimensional topological $\mathrm{pin}^-$ bordism gives rise to a $\mathbb{Z}/2\mathbb{Z}$ classification. In this sense, it looks that the $\mathrm E_8$-manifold should be a manifold generator of the $\mathbb{Z}/2\mathbb{Z}$ classification of both topological non-orientable $\mathrm{pin}^+$ and $\mathrm{pin}^-$ bordism, but not the orientable topological spin bordism. Then, should $\mathrm E_8$-manifold be orientable or not?