To start with, consider some abstract $3$-dimensional simplicial complex $\Delta$ representing a manifold without boundary, for simplicity. Then, there is this well-known construction of the "(Poincare) dual complex". By definition, it defined to be the complex obtained as follows:
- For every tetrahedron ($3$-simplex), the dual complex has a vertex ($0$-cell) in its barycentre.
- If two tetrahedra are glued together along a triangle ($2$-simplex), then we connect the two vertices of the corresponding dual complex by an edge ($1$-cell). Note that the edge intersects the corresponging triangle at exactly one point.
- Porceeding this procedure, we define faces ($2$-cells) of the dual complex to be in one-to-one correspondence with the edges ($1$-simplices) of the triangulation.
To sum up, the dual complex is a (in general non-simplicial) cellular complex $\Delta^{\ast}$, where to every $k$-cell there is a corresponding $(3-k)$-simplex in $\Delta$. Of course, this definition can be extended to higher-dimensions, where every $k$-cell of the dual complex is in one-to-one correspondence to a $(d-k)$-simplex.
Now, it is not too hard to see that the dual complex of some manifold triangulation represents the same manifold. However, it seems to me that this is no longer true if $\Delta$ is a "pseudomanifold". A pseudomanifold is defined to be a (geometric realization of a) strongly-connected, non-branching and pure simplicial complex. More geometrically, one can think of pseudomanifolds as topologies, which are manifolds in most of their points, but can fail to be locally Eucilidean in some isolated points, which are singular. As an example, consider the topology obtained by identifying to distinct points on the $2$-sphere, i.e. the pinched torus, which is an example of a (non-normal) $2$-dimensional pseudomanifold.
Now, as an example, let $\Delta$ be a $3$-dimensional pseudomanifold. Then, we can calculate its Euler-characteristic as $$\chi(\Delta)=\Delta_{0}-\Delta_{1}+\Delta_{2}-\Delta_{3},$$ where $\Delta_{i}$ denotes the number of $i$-simplices. By definition, the corresponding dual complex has Euler characteristic $$\chi(\Delta^{\ast})=\Delta_{3}-\Delta_{2}+\Delta_{1}-\Delta_{0}=-\chi(\Delta).$$ Now, the point is, if $\Delta$ is a manifold, then $\chi(\Delta)=0$ (since every odd-dimensional compact manifold has Euler characteristic $0$), however, if $\Delta$ is a pseudomanifold, then its Euler charactersitic can have a non-vanishing number. It is not too hard to construct examples. Since every pseudomanifold is in particular a CW-complex and we know that the Euler characteristic is a topological invariant, the fact that $\chi(\Delta)\neq\chi(\Delta^{\ast})$ tells us that $\Delta^{\ast}$ cannot represent the same topology as $\Delta$.
I found this observation rather weird and hence, I wanted to ask if the argument from above is true, or if it contains some error. For example, I have seen the statement that the dual complex represents the same topology as the corresponding simplicial complex in a number of papers on quantum gravity (in a context where they also allow for pseudomanifolds), so I am curious if I am overlooking something.
