Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A procedure is described in section 3.2 of this article: $\pi_1(C\setminus D)$ is generated by the $n-1$ cells in $C$ and the relations are given by the $n-2$ cells in $C\setminus D$. This fact is not proven there and regarded as "well known".

Can you give me a reference? or a proof?

EDIT: Since the formulation of the article is false (see comment below), I added the hypothesis that $C$ is a differentiable manifold.