Why any $(n -1)$-face of a $n$-simplex is the face of exactly one other $n$-simplex?

Question

I think it's possible to be contained in odd numbers of other $n$-simplexes, for example we may identify the edges of two cube's boundaries.

Answer 1

Although what you say is possible in a general simplicial complex, it is not possible in a simplicial complex which is a triangulation of an $n$-manifold.

To see why, use local homology calculations.

First, for any $n$-dimensional manifold $M$ and for any point $x \in M$, the $n$-dimensional local relative homology group $H_n(M,M-x;\mathbb{Z})$ is isomorphic to $\mathbb{Z}$.

However, for any $n$-dimensional simplicial complex $X$, and for any $n-1$ dimensional simplex $\sigma$ which is a face of exactly $k$ simplices of dimension $n$, and for every point $x$ in the interior of $\sigma$, the group $H_n(X,X-x;\mathbb{Z})$ is isomorphic to $\mathbb{Z}^{k-1}$.

Putting these together, if $X$ is an $n$-manifold then $k-1=1$ so $k=2$.

On an intuitive level, what this is saying is that a $n-1$ dimensional plane in $\mathbb{R}^n$ locally separates into two pieces, and that's what's happening locally near a point in the interior of an $n-1$ simplex in an $n$-manifold.