Question: If $X_4$ is a non-triangulable manifold,
can $X_4 \times S^1$, $X_4 \times I^1$, or $X_4 \times \mathbb{R}^1$ be a triangulable manifold?
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.
