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.

Hatcher 0.21
Figure 1



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