Is the surface of a 3d cube homeomorphic to the 2-sphere

Question

Is the surface of a three dimensional cube i.e. $[0,1]^3$ (surface is like a hardboard box, I don't know if it has a symbol to represent) homeomorphic to $S^2$. If yes, is it diffeomorphic to $S^2$ too?.

My conception is that we can put a cube inside a spherical balloon and somehow deflate the air in the balloon to get it plastered on the cube, so this seems like a continuous mapping of $S^2$ to the surface of a cube, but I am not sure whether it is continuous or not. Also if it is continuous is that a diffeomorphism also?

Answer 1

I think the main issue here is what exactly you mean by the cube.

The cube, as a subset of $\mathbb{R}^3$ with the subspace topology, is a topological space. The sphere, also with the subspace topology inherited from $\mathbb{R}^3$, is also a topological space. So at this point it makes sense to ask whether they are homeomorphic, and yes they are. To see this you can use your 'deflating' argument.

In order to ask for a diffeomorphism between these spaces you need specify what differentiating means on them, i.e. you need to specify a manifold structure. While $\mathbb{S}^2\subset\mathbb{R}^3$ (more or less canonically) inherits a such a structure to be a submanifold of the ambient $\mathbb{R}^3$, this is not true for the cube. In pictures: You cannot find a continuous choice of tangent spaces in the edges/corners. So as long as you do not specify a manifold structure it does not make sense to ask for a diffeomorphism.

You can save this as follows. Whenever you have a homeomorphism $\phi:X\to M$ from some topological space $X$ into a manifold $M$ you can pull back the manifold structure through $\phi$ to turn $X$ into a manifold and, w.r.t. this manifold structure on $X$, $\phi$ is now a diffeomorphism by construction. So, using the 'deflation' homeomorphism between cube and sphere one can endow the cube with a manifold structure such that the cube and the sphere are diffeomorphic. But note that the cube is now not a submanifold of $\mathbb{R}^3$, in particular, the tangent space in a edge/corner corner point $p$ is not a subspace of the tangent space $T_p\mathbb{R}^3$ of the ambient space, but some abstract space which you have 'glued into' the problematic points.

So concluding, if you specify the 'correct' notion of what diffeomorphic means, then they are diffeomorphic by construction, but in a canonical sense of subsets/submanifolds of $\mathbb{R}^3$ the question for a diffeomorphism makes no sense.

Answer 2

They are homeomorphic suspaces of $\mathbb R^3$. A formal proof can be found in my answer to Homeomorphism between a semicircle and a line where I considered the surface $C^2$ of the cube $[-1,1]^3$. The sphere $S^2$ sits inside $C^2$ and $C^2$ is deflated. You can replace $[-1,1]^3$ by any other cube $[a,b]^3$; note that $[-1,1]^3$ and $[a,b]^3$ are obviously homeomorphic which gives a homeomorphism between the surfaces.

Are they diffeomorphic? This question does not make sense unless we give $S^2$ and $C^2$ smooth structures. The sphere is a smooth submanifold of $\mathbb R^3$ and thus "the natural way" to endow it with a smooth structure is to take that inherited from $\mathbb R^3$. But $C^2$ is not a smooth submanifold of $\mathbb R^3$ (it has edges and corners), thus it does not have a completely natural smooth structure. However, you can take a homeomorphism $h : S^2 \to C^2$ to transfer the smooth structure of $S^2$ to a smooth structure on $C^2$, and this trivially makes $h$ a diffeomorphism.