Hatcher 0.21

Hatcher, Algebraic Topology, Chapter 0

21. If $X$ 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 $X$ is homotopy equivalent to a wedge sum of $S^1$‘s and $S^2$‘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.

Figure 1

$\square$