# Pseudomanifolds

An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that

(i) every simplex is a face of an $n$-simplex

(ii) every $(n-1)$-simplex is a face of exactly two $n$-simplices

(iii) Given any two $n$-simplices $\sigma, \tau \in X$ there is a sequence of $n$-simplices $\sigma_0 = \sigma, \ldots, \sigma_k = \tau$ such that $\sigma_i \cap \sigma_{i+1}$ is an $(n-1)$-dimensional simplex for each $0 \leq i \leq k-1$.

These conditions imply that the polyhedron of a pseudomanifold is path-connected. Is it true that if a finite simplicial complex $X$ satisfies (i) and (ii) and has a path-connected polyhedron then it satisfies (iii)?

• Just as an idea for a proof that its true, suppose you take a path connecting vertices for two simplices $\sigma$ and $\tau$. This path is homotopic to a path which transverses only simplex edges and vertices. I believe that if you can prove that the star of any vertex (the simplices that intersect that vertex) is a Pseudomanifold then you can prove that there exists a chain between $\sigma$ and $\tau$ as you describe. – Eric Haengel Feb 18 '13 at 18:54
• "Simplicial stuff"? – Neal Feb 18 '13 at 20:40
• @Neal Yeah, that tag rubs me the wrong way every time I see it. This is not OP's decision, the tag has been used for a while. It was criticized in the past. I personally feel that the simple (simplicial-) would be better. – user53153 Feb 18 '13 at 22:39

Counterexample with $n=2$: let $X$ be the union of two hollow tetrahedra joined at a vertex. Here (i) and (ii) hold; (iii) fails, yet $X$ is path-connected.