Timeline for Strong Whitney embedding theorem for non-compact manifolds
Current License: CC BY-SA 3.0
18 events
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| May 26, 2013 at 20:22 | comment | added | Ricardo Andrade | I thus reiterate my previous statement: an injective continuous map $f$ between two manifolds is proper in Whitney's sense if and only if $f$ is a homeomorphism onto its image. | |
| May 26, 2013 at 20:13 | comment | added | Ricardo Andrade | @Liviu: I strongly believe that the passage you quote refers to the function $f^{-1}$, not to the operation of taking inverse image of subsets of the target. In fact, your quote from section IV.A.1 of Whitney's book skips/omits an important part which clarifies what Whitney means. I will now give the full quote: "It is easy to see that a one-one mapping $f$ is proper if and only if the inverse $f^{-1}$ is continuous in $f(M)$, or, if and only if $f^{-1}$ carries compact sets into compact sets." That sentence is valid only when the second $f^{-1}$ also denotes the inverse function. | |
| May 26, 2013 at 11:07 | comment | added | Liviu Nicolaescu | I quote Whitney, in IV.A.1, just before he defines the notion of imbedding "It is easy to see that a $1-1$ mapping $f$ is proper if and only if [...] $f^{-1}$ carries compact sets into compact sets." | |
| May 25, 2013 at 11:57 | comment | added | Ricardo Andrade | Also, let me make a curious note on evolution of terminology. When Whitney says that $f:X\to Y$ is "proper", he does not mean that: (#) the inverse image of a compact subspace of $Y$ by $f$ are compact (the current usual meaning of proper map). In fact, if $f$ is injective, Whitney's notion of "proper" just means that $f$ is a homeomorphism onto its image. A map $f$ verifying (#) would actually be called by Whitney a "mapping without limit set". See, for example, the discussion preceding the statement of theorem IV.1A in "Geometric integration theory". | |
| May 25, 2013 at 11:38 | comment | added | Ricardo Andrade | @Liviu: I saw theorem 1A in chapter IV, and its proof for the non-compact case in section IV.A.7. This appears to not be the situation I am asking about, since the theorem only gives an embedding of $n$-dimensional manifolds into $\mathbb{R}^{2n+1}$, while I am asking about embeddings into $\mathbb{R}^{2n}$. | |
| May 25, 2013 at 11:07 | comment | added | Liviu Nicolaescu | The noncompact case is discussed in IV.A.7 of Whitney's book. Again, Whitney's definition of an embedding includes the properness of the map. | |
| May 25, 2013 at 11:03 | comment | added | Liviu Nicolaescu | The image of a proper embedding is a closed subset. | |
| May 25, 2013 at 11:01 | comment | added | Liviu Nicolaescu | Whitney defines an embedding is a 1-1 proper map. | |
| May 25, 2013 at 9:04 | vote | accept | Ricardo Andrade | ||
| May 24, 2013 at 23:05 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| May 24, 2013 at 22:57 | history | edited | Ricardo Andrade |
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| May 24, 2013 at 22:54 | comment | added | Ricardo Andrade | @Liviu: The only related result I could quickly find in Whitney's "Geometric integration theory" is his weak embedding theorem concerning embeddings of a $n$-dimensional manifold into $\mathbb{R}^{2n+1}$. This is much more standard than the strong embedding theorem I am asking about, concerning embeddings into $\mathbb{R}^{2n}$. Did I happen to miss this result in the book you mentioned? | |
| May 24, 2013 at 22:45 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| May 24, 2013 at 11:52 | comment | added | Liviu Nicolaescu | Have a look at Whitney's "Geometric integration theory". It contains a proof of his embedding theorem which shows that the image of the embedding is closed. | |
| May 24, 2013 at 10:24 | answer | added | Ryan Budney | timeline score: 26 | |
| May 24, 2013 at 10:22 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| May 24, 2013 at 10:15 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| May 24, 2013 at 10:04 | history | asked | Ricardo Andrade | CC BY-SA 3.0 |