What is exactly a pseudo-manifold? I read the Wikipedia article, and according to it a pseudo-manifold is a finite simplicial complex with the following conditions hold:
- (pure) $X = |K|$ is the union of all $n$-simplices.
- Every $(n – 1)$-simplex is a face of exactly two $n$-simplices for $n > 1$.
- For every pair of $n$-simplices $σ, σ' \in K$, there is a sequence of $n$-simplices $σ = σ_0, σ_1, …, σ_k = σ'$ such that the intersection $σ_i ∩ σ_{i+1}$ is an $(n − 1)$-simplex for all $i$. [1]
...but condition $3$ it is still not completely clear to me. Even more confusing, an another definition says
A regular $(d-1)$-connected $d$-complex in which all $(d-1)$-simplices are manifold simplices is called a combinatorial pseudomanifold. [2]
Are combinatorial pseudomanifold and pseudomanifold two different things?
Regarding quasi-manifolds, it's much harder to find a proper definition. The only one I found is
A regular $h$-complex in which the star of every vertex is $(h-1)$-manifold-connected is called an initial quasi-manifold. [2]
How is a quasi-manifold different from a pseudo-manifold?