If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$ always be a triangulable manifold?

Question

Let $X_d$ be a $d$-manifold which is not a triangulable manifold but only a topological manifold.

Question:

• Is this true that $X_d \times T^k$:

1. The $X_d \times T^k$ can always be triangulable?

2. If not true, can $X_d \times T^k$ sometimes be triangulable? under what criteria? (for example, for a certain dimension $d$? for a certain bound on $k$? or when $X_d$ has a certain structure?)

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Other info:

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

• Any orientable 5-manifold is triangulable.