Let $S$ be a genus two surface embedded into $\mathbb{R}^{3}$. Let $H$ be the solid handlebody bounded by $S$. Find a cell structure for $H$. As well, find a simplicial complex homeomorphic to $H$.
I'm not sure on how to start this problem. First off, how could one visualize the handlebody $H$ as a subspace of $\mathbb{R}^3$? As well, I think I know how to give a cell structure for $S$ (a cell complex with the representation $\langle a,b,c,d | [a,b][c,d]\rangle$, however I'm not sure on how to use this fact to obtain a cell structure for $H$.
For the other question, I know that a simplicial complex is a cell complex formed by joining simplices along subsimplices where every closed $n$-cell is a closed $n$-simplex, and that the intersection of two simplices is a simplex. However, I'm not so sure if I understand this definition. What does one mean with a closed $n$-cell and a closed $n$-simplex? Then, how would I be able to use this definition to find the simplicial complex for $H$?
Any help would be appreciated.