In differential topology we have a chain of strict inclusions $$\mathsf{Diff} \subsetneq \mathsf{PL} \subsetneq \mathsf{Triang} \subsetneq \mathsf{Top}$$ among the classes of smooth, PL, triangulable and topological manifolds, respectively.
Now consider the class $\mathsf{Handle}$ of topological manifolds that admit a handle decomposition.
Question: Does $\mathsf{Handle}$ fit in the previous sequence? (and if it does, where?)
Any PL manifold admits a handle decomposition and for that (e.g. see Rourke, Sanderson, Introduction to piecewise-linear topology) one can argue with a triangulation coming from the PL structure. Of course, this will be a combinatorial triangulation, but there are triangulable manifolds that do not admit any PL structure.
It feels to me that $\mathsf{Handle}$ and $\mathsf{Triang}$ are not contained one in the other, but I don't know any counterexamples.