For Topology questions, often a good start is to ask yourself "What's the picture?"

(2/n)

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(4/n)

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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)

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So first up is looking for this decomposition.

(7/n)

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(13/n)

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(14/n)

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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) ✌️😁

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