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May 24, 2020 at 12:59 comment added Qfwfq As noticed in the other comment, punctured disk and circle are not diffeomorphic. Also, the first Betti number of the punctured disk is not zero (if I understand correctly, what you denote by $D^1$ is the unit disk).
May 24, 2020 at 12:34 comment added Ben McKay The notation $D^1$ for the unit disk is not very pleasant. Perhaps just $D$. Superscripts are usually for dimensions. I don't agree that the punctured unit disk is diffeomorphic to the circle.
Dec 1, 2018 at 1:45 history answered Morino_Hikari CC BY-SA 4.0