Fundamental group of the complement of cell subcomplexes 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.
 A: Let $B^k$ be the standard $k$ dimensional ball.  An $n$-dimensional $k$-handle is a copy of $B^k \times B^{n - k}$.  The boundary of a $k$-handle is 

$((\partial B^k) \times B^{n - k}) \cup (B^k \times (\partial B^{n - k}))$

We call the first part of the boundary the attaching region; this will glue "down" to smaller (in $k$) handles.  The second half of the boundary is attached to by larger handles. 
Suppose that $C$ is a compact connected $n$-manifold without boundary.  Suppose that $C$ is equipped with a handle structure where 


*

*the attaching maps are very nice,

*there is exactly one zero-handle, and 

*there is exactly one $n$-handle.


Then we can compute the fundamental group of $C$ in two ways.  We can use the zero-, one-, and two-handles. Or we can dualise and use the $n$-, $(n-1)$-, and $(n-2)$-handles.  As suggested in the original post, after dualising, the $n$-handle gives the base point, the $(n-1)$-handles give generators, and the $(n-2)$-handles give relations.  If you remove $D$, made up of $n-2$ and smaller handles, then the dual presentation of the fundamental group still holds (but perhaps with fewer relations). 
