The CW structure of has one cell in each dimension for k = 0, 1, and 2 such that .
The attaching map for is the 2-sheeted covering projection for k=1,2, where is a point and is a circle.
Computing a boundary map is not trivial. A good reference for this is Hatcher's "Algebraic Topology" p 137-148, especially Example 2.42.
The resulting cellular chain complex is as follows:
Now the homology group of is
and for .