I'm working out the homology groups of the one point union of the circle and the sphere.
The obvious CW complex is 1 0-cell (the point where the circle and sphere touch), 1 1-cell (the circle) and 1 2-cell (the sphere).
So we get a sequence
0 -> Z -> Z -> Z -> 0
d_2 d_1 d_0
H_{0}=Z
The first homology group, I think is Z, because the kernel of d_1 is Z and d_2 has degree 0 and so im d_2 = 0.
Then the second homology group, ker(d_2)/Z=0 as ker(d_2) + im d_2 = Z, and the map from 0->Z has image Z.
Is this correct?