Given a triangulated homology manifold $X$, the dual block decomposition is defined by setting, for each simplex $\sigma$ of $X$, the block $\overline{D}(\sigma)$ to be the union of all simplices of $\text{sd}\ X$ of the form $\hat{\sigma}_1 \cdots \hat{\sigma}_k$ (with $\sigma_1 \succ \sigma_2 \succ \cdots \succ \sigma_k$, the symbol $\succ$ meaning 'has as proper face' and hat meaning barycenter) where $\sigma_k = \sigma$ (the open block $D(\sigma)$ is the union of all open such simplices). This is the approach used in Munkres's Elements of Algebraic Topology. Equivalently, for each vertex $v$ one has $\overline{D}(v)$ the closed star of $v$ in $\text{sd}\ X$; for an edge $vw$ one has $\overline{D}(vw) = \overline{D}(v) \cap \overline{D}(w)$ etc. This dual decomposition can be used to establish Poincaré duality.
When is this dual block decomposition a CW decomposition? I'm under the impression this is guaranteed if the triangulation of $X$ is PL. Indeed, for a vertex $v$, one has $\overline{D}(v) = v * \text{Lk}(v, \text{sd}\ X)$, which is a cell since $\text{Lk}(s, \text{sd}\ X)$ is a sphere. For a general simplex $\sigma$, though, all I know is that $\text{Lk}(\hat{\sigma}, \text{sd}\ X) = \dot{D}(\sigma) * \text{sd}(\text{Bd}\ \sigma)$, where $\dot{D}(\sigma) = \overline{D}(\sigma) - D(\sigma)$ (and so $\overline{D}(\sigma) = \dot{D}(\sigma) * \hat{\sigma}$) -- I'm not sure if this guarantees $\dot{D}(\sigma)$ is a sphere. Furthermore one might wish to worry about attaching maps, though that seems benign to me.
What's an example of a triangulable manifold where the dual block decomposition is not a CW decomposition?
