What is the $CW$ structure of connected sum of $Rp^2$ I am wondering how to describe a $CW$ structure of connected sum of $Rp^2$ to calculate its homology. To be honest, I have no idea about what should the connected sum of $Rp^2$ look like. Can some give some suggestion?
 A: The disk has a CW structure $e^0 \cup e^0 \cup e^1 \cup e^1 \cup e^2$ with the 0- and 1-cells on the boundary. By identifying antipodal points on the boundary we see that $RP^2$ has a CW structure with cells $e^0 \cup e^1 \cup e^2$ (the two 0-cells were identified and the two 1-cells were identified). 
The connect sum is obtained by removing a disc of each and gluing along the boundary of these disks. We may as well take the discs to be contained in the interior of $e^2$. To accommodate for this we have to change the CW structure on $RP^2$ slightly. Just as before we can work out the CW structure on the unit disk first and this will give a CW structure on $RP^2$.
We still have the cells $e^0 \cup e^0 \cup e^1 \cup e^1$ on the equator. We want 1-cells so looking down it looks like a pokeball.

The CW structure here should have four 0-cells, six 1-cells, and three 2-cells. I think its pretty clear what these are (keeping in mind that this is an inductive construction). The associated CW structure on $RP^2$ just has one less 0- and 1-cells. 
We can form the connect sum of two $RP^2$ by removing the middle 2-cell of each and gluing along the boundary, i.e. the middle circles. Now what is its CW structure? Its constructed inductively by attaching 0-cells then 1-cells and then 2-cells. All you have to do at this point is keep in mind that, for instance, you don't need all the 0-cells from each because the two pairs on the middle circle of each $RP^2\setminus e^2_{middle}$ end up being identified. I'll leave it to you what the characteristic maps are more precisely.
