Return to Question

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).

Edit: If I understand it well, Misha Kapovich points out in his comment below that Robion Kirby told him that the answer to question (2) above is positive, in every dimension $n\geq 5$ (and thus in every dimension $n\geq 4$, with the difference that for $n\geq 5$ no such sets can be homeomorphic to $\mathbb{R}^n$). As (2) implies (1) this would settle the question.

It would be interesting to know if other mathematicians heard about these examples of Kirby, and specially if some of these open sets admit a simple description up to homeomorphism.

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).

Edit: If I understand it well, Misha Kapovich points out in his comment below that Robion Kirby told him that the answer to question (2) above is positive, in every dimension $n\geq 5$ (and thus in every dimension $n\geq 4$, with the difference that for $n\geq 5$ no such sets can be homeomorphic to $\mathbb{R}^n$). As (2) implies (1) this would settle the question.

It would be interesting to know if other mathematicians heard about these examples of Kirby. From the topological point of view (i.e. up to homeomorphism), what are these sets?

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).

Edit: If I understand it well, Misha Kapovich points out in his comment below that Robion Kirby told him that the answer to question (2) above is positive, in every dimension $n\geq 5$ (and thus in every dimension $n\geq 4$, with the difference that for $n\geq 5$ no such sets can be homeomorphic to $\mathbb{R}^n$). As (2) implies (1) this would settle the question.

It would be interesting to know if other mathematicians heard about these examples of Kirby, and specially if some of these open sets admit a simple description up to homeomorphism.

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).

Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4.

Questions: In dimension $n\geq 5$,

1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?

2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?

Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism?

(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).