Oriented Topological Pseudomanifolds and their Homology

Question

Let $X$ be a topological space endowed with a triangulation $K$. We say that $X$ is a $n$-dimensional pseudomanifold, if

1. $X$ is the union of all $n$-simplices
2. Every $(n-1)$-simplex is a face of exactle two $n$-simplices for $n>1$
3. For every pair of $n$-simplices $\sigma$ and $\tau$ in $K$, there is a sequence of $n$-simplices $\sigma=\sigma_0, \sigma_1, \ldots , \sigma_k=\tau$ such that the intersection $\sigma_j \cap \sigma_{j+1}$ is a $(n-1)$-simplex for all $j$.

Let $X$ be a $n$-dimensional pseudomanifold. I want to prove that the following are equivalent:

• For all $(n-1)$-simplex $\sigma$, all the simplices of which $\sigma$ is a face, induces contrary orientation on $\sigma$.
• $H_n(K)\not=0$

Here orientation means an ordering of the vertices.

Answer 1

Hint: Let $[X]$ be the chain equal to the sum of all $n$-simplexes, show that in fact $[X]$ is a cycle.