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Basic to all mathematics is the notion-here used quite informally-of a set with structure. For every type of structure there is a notion of equivalence (or isomorphism)-a one-to-one onto function that, in an appropriate sense, preserves the structure. A particular type of structure defines a branch of mathematics: the study of those concepts preserved by equivalence. For example, a group is a set furnished with the structure group operation. The notion of equivalence is the usual notion of isomorphism of groups. (Semi-Riemannian geometry with applications to Relativity, by Barrett O’neill)

Consider following table:

enter image description here

I know if $f:X\to Y$ be a one-to-one onto function, then two sets $X$ and $Y$ have the same cardinality and it is sufficient for me to visualize the equivalence in this case.

If $f:X\to Y$ be a homeomorphism, then one can deform the topological space $X$ to the topological space $Y$ without cutting and gluing, it is sufficient for me to visualize the equivalence in this case. If a coffee cup and a donut are given to me, I can realize from their shapes that they are homeomorphic.

enter image description here

If $f:X\to Y$ be an isometry, then one can coincide the semi-riemannian manifold $X$ to the semi-riemannian manifold $Y$, it is sufficient for me to visualize the equivalence in this case.

Question 1: Is the above statement true?

Question 2: I have no idea to visualize the equivalence in the Manifold theory case, diffeomorphism. Can someone help me? Can I judge about the equivalence of the two manifolds from their shapes in this case?


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Have you ever checked this in Spivak? – Babak S. Aug 2 '13 at 20:15

Regarding Q1, by and large the answer is no. Many homeomorphisms can be viewed this way. But many also can not. The above description of homeomorphisms is actually far more appropriate for diffeomorphisms than homeomorphisms. You can make the statement quite precise, using the isotopy extension theorem. I'll comment on that after Q2.

Q2: I think you are wrong here. The smooth case is far easier!

Okay, so the isotopy extension theorem is the key. Here is how it goes. Given a smooth manifold $M$ of dimension $m$, you can embed it (smoothly!) in a Euclidean space. You can embed it in $\mathbb R^{2m+1}$ via a general position argument of Whitney's. But you can go much further. If you embed your manifold $M$ into $\mathbb R^{2m+3}$, it turns out that that embedding is unique up to ambient isotopy. i.e. for any two embeddings, there is a 1-parameter family of diffeomorphisms of the euclidean space that carries the first embedding to the second. This is the precise "deformation" that you are talking about above. The fact that the embedding is unique up to isotopy is an old theorem of Whitney's. It boils down to an argument where you deform the straight-line homotopy between the two maps into an isotopy, and then use the isotopy-extension theorem to extend that isotopy to an ambient isotopy (sometimes called a diffeotopy). In a sense diffeomorphisms are more restrictive than what you are describing as diffeomorphisms do not allow violation of the local linear structure -- you can't turn a circle into a square using smooth isotopies.

Provided your spaces are "nice enough", homeomorphisms can frequently be chosen to be similar-enough to diffeomorphisms that you can make something like the above work out, and your "deformation" outlook holds. But there are spaces where homeomorphisms between them are "ugly" and there's no such machinery like the above. You might want to google-around and look at Cantor sets in $\mathbb R$, $\mathbb R^2$ and $\mathbb R^3$. They are all homeomorphic but there is no deformation between them (well, much of the time). And it doesn't get any better by going up to higher dimensions. Or course, the precise answer depends on what you mean by "deform" in this context -- when dealing with topological spaces that's a slippery subject. I'm using the "homeotopy" idea, i.e. a continuous 1-parameter family of homeomorphisms of the ambient space carrying one object to the other.

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