Why do we have $H_2(\mathbb P^1) \cong H_1(\mathbb A^1\setminus \{0\})$?

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This question comes from Matt E's answer in

What is the intuition behind the concept of Tate twists?

He remarks that $H_2(\mathbb P^1) \cong H_1(\mathbb A^1\setminus \{0\})$, but I don't see where this comes from. Could someone give more detail?