Let $M$ be a connected differentiable manifold. How can one describe those closed subsets $A$ of $M$ such that there is a diffeomorphism $\varphi:( M - A )\to M$ ?
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1$\begingroup$ Do you have any reason to expect there to be a reasonable such description? Is there one in the case of e.g. $M=\mathbb R^3$? $\endgroup$– WojowuCommented Jan 21, 2021 at 19:56
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2$\begingroup$ Wild embeddings (like Alexander horned sphere) show that this can't be an intrinsic property of $A$. The $h$-cobordism theorem implies a non-trivial sufficient criterion: if $M$ and $M \setminus A$ are the interiors of compact simply connected $d$-manifolds with simply connected boundary and $d \geq 6$, then it suffices that $\check{H}^*_c(A) = 0$. $\endgroup$– user171227Commented Jan 21, 2021 at 21:19
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$\begingroup$ Trivial comment: when $M$ is closed (compact no boundary) then this happens if and only if $A = \emptyset$. $\endgroup$– user171227Commented Jan 21, 2021 at 21:41
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