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.

Question: If $X_4$ is a non-triangulable topological (TOP) 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?

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

Note that we have smooth (DIFF) ⊂ PL ⊂ triangulable ⊂ TOP.

(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 more specific version of the previous one focusing on $d=4$ only.