The space X is the real projective plane, , which is the quotient space , where -x is the antipodal point of x for each .

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 .