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Sufficient condition for the union of two submanifolds to be a submanifold

In the meantime, a very similar question of mine has been answered here https://math.stackexchange.com/a/4642619/857154 , which answers these questions aswell. Moishe Kohan has provided a counterexamp …
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Sufficient condition for the union of two submanifolds to be a submanifold

I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that $(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and $(ii)$ $\overline{S …
Euler Characteristic's user avatar