Topology of 2-manifold I'm reading an old paper from Roberts and Steenrod called "Monotone transformations of two-dimensional manifolds". The problem that I'm facing is that the language used by the authors isn't modern and I'm not an expecialist in topology, so I could'n translate the statements to modern topology language. The Lemma that I'm especially confused is:
$\textbf{Lemma 1}$: Let $M$ be a compact 2-manifold without boundary, $g$ a continuum on $M$ and $D$ a component of $M-g$. Then $D$ contains a finite closed connected complex $K$ whose $mod\; 2$ boundary is the sum of a finite number $\Gamma_1, \cdots, \Gamma_s$ of pairwise disjoint simple closed curves, and $D-K = H_1 + \cdots + H_s$, where the $H_i$ are mutually exclusive open cylinders and the boundary of $H_i$ is $\Gamma_i$ plus a subset of $g$. 
Could anyone help me to understand this Lemma, especially the part  "$D$ contains a finite closed connected complex $K$ whose $mod\; 2$ boundary is the sum of a finite number of pairwise disjoint simple closed curves" ?
 A: A complex $K$ in this context means a simplicial complex embedded in $M$. 
Connectedness of $K$ you understand already, I'm sure. 
The phrase "finite closed" is a bit redundant, because I'm pretty sure that "closed" just means that $K$ is a closed subset of $M$, and therefore $K$ is compact, and therefore $K$ has finitely many simplices, which is exactly what "finite" means. 
To understand the "mod 2 boundary" of $K$, using the simplicial structure on $K$ you can define the simplicial chain complex of $K$ with $\mathbb Z / 2 \mathbb Z$ coefficients:
$$0 \mapsto C_2(K;\mathbb Z / 2 \mathbb Z) \mapsto C_1(K;\mathbb Z / 2 \mathbb Z) \mapsto C_0(K;\mathbb Z / 2 \mathbb Z) \mapsto 0
$$
In $C_2(K;\mathbb Z / 2 \mathbb Z)$, there is a unique 2-chain that I'll denote $D_K$ whose coefficient on each 2-simplex of $K$ is the nonzero element of the coefficient group $\mathbb Z / 2 \mathbb Z$. Consider its boundary $\partial D_K$, an element of the group of 1-cycles $Z_1(K;\mathbb Z / 2 \mathbb Z)$. Notice that the support of $D_K$ is the union of all 1-simplices which are on the boundary of exactly one 2-simplex of $K$ (in general, for any 1-simplex $\tau$ of $K$, $\tau$ is on the boundary of exactly zero, or one, or two 2-simplices of $K$; this is true because $K$ is embedded in a surface).
The meaning of the first part of the conclusion is that one can write
$$\partial D_K = \Gamma_1 + .... + \Gamma_s
$$
where each $\Gamma_i$ is an element of $Z_1(K;\mathbb Z / 2 \mathbb Z)$ whose support is a simple closed curve, and hence by abusing terminology one can identify $\Gamma_i$ with that simple closed curve. Furthermore, these simple closed curves are pairwise disjoint, and then further properties of them are described.
One could state a stronger and more succinct conclusion if one wished, by saying that $D$ contains a compact surface $\Sigma$ whose boundary is a finite union of simple closed curves $\partial\Sigma = \Gamma_1 \cup \cdots \cup \Gamma_s$, with further properties as described. One would just take $\Sigma$ to be the union of the 2-simplices of $K$; equivalently, one would throw away the interiors of the 1-simplices of $K$ that are not on the boundary of any 2-simplex of $K$. 
