Hatcher, Algebraic Topology, Chapter 0
21. If is a connected Hausdorff space that is a union of a finite number of 2-spheres, any two of which intersect in at most one point, show that is homotopy equivalent to a wedge sum of ‘s and ‘s.
Proof. This follows immediately by induction on the procedure outlined in Example 0.9 of Hatcher’s book (pg 12), which makes use of the diagram in Figure 1 below.