For Topology questions, often a good start is to ask yourself "What's the picture?"
(2/n)
(4/n)
Whenever you have these "adjunction" spaces given by gluing, using the Seifert van-Kampen theorem to compute π_1 is usually a good bet.
(6/n)
So first up is looking for this decomposition.
(7/n)
(13/n)
(14/n)
1. How can you modify the original problem so we get nontrivial π_1 in the pieces?
2. Is this decomposition enough to use Mayer-Vietoris to compute the homology H_*(X)?
3. Where did the dimension matter? Does this work for S^{n-1} in S^n?
(16/16) ✌️😁