Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ necessarily separates $M$ (i.e., represents the trivial class in $H_2(M, \partial M;\mathbb{Z})$)?
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3$\begingroup$ No. Choose a 2-sphere $S=S^2\times\{pt\}$ which intersects $B$ in a disk. Then $D=S\setminus B$ is a disk which does not separate $M$. $\endgroup$– Josh HowieCommented Sep 29, 2020 at 2:53
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1$\begingroup$ Simple! Thank you, Josh. $\endgroup$– Eduardo LongaCommented Sep 29, 2020 at 3:01
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