TITLE: Taking the connected sum of four closed disks... What do we get?
QUESTION [6 upvotes]: If we take the connected sum of four closed disks $S = 4 \mathbb{\overline{D}} = \mathbb{\overline{D}} \# \mathbb{\overline{D}} \# \mathbb{\overline{D}} \#  \mathbb{\overline{D}}$, what does $S$ look like and how do we describe the boundary? (This has been resolved) Is it just a $2$-sphere, since $ \mathbb{\overline{D}} \# \mathbb{\overline{D}}  = \mathbb{S}^2$ and $ \mathbb{S}^2 \# \mathbb{S}^2 = \mathbb{S}^2?$
How do we write down a plane model for this surface? If $S = 4 \mathbb{\overline{D}}$ is an annulus, would it just have the same plane model as the cylinder? (Since they are homeomorphic)

REPLY [5 votes]: First of all, as Mike Miller points out in his comment, $D^2 \# D^2 \approx S_1 \times [0,1] $ which is just a cylinder. So what you get is the connected sum of two cylinders. 
In terms of writing down the planar model for this surface, this question of mine will be useful: What is the word associated with the connected sum of two surfaces with boundary?. The current answer is right for manifolds without boundary but not for manifolds with boundary. I found the answer for manifolds with boundary and I'll post it soon, but the result is that the word associated to the connected sum of surfaces with boundary (with associated words $p_1$ and $p_2$) is $c^{-1}p_1cp_2$. 
The cylinder has as an associated word $aba^{-1}c$, so the word associated to the connected sum of two cylinders is:
$$g^{-1}aba^{-1}cgded^{-1}f$$
With this you can create the planar model and classify the surface using the classification of compact manifolds.