Question: If $X_4$ is a non-triangulable manifold, 

1. can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold? 

2. can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a PL manifold? 

(If $X_4$ spin or non-spin manifold makes a difference for the answer, then we should discuss the differences.) The $I^1$ means a 1-dimensional finite internal.

---

Let $X_4$ be a $4$-manifold which is *NOT* a triangulable manifold but only a topological manifold. 





----

Other warm-up info:
--

- If $X_4$ is the non-triangulable Freedman's E8 topological manifold, then $X_{4+𝑘}=X_4\times T^𝑘$ is triangulable, but not piecewise linear (PL). 

- Any orientable 5-manifold is triangulable.

This question is a simplified version of [the previous one](https://mathoverflow.net/q/385189/27004).