Timeline for Codimension zero embeddings and diffeomorphism groups
Current License: CC BY-SA 3.0
15 events
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| Dec 16, 2013 at 23:24 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Dec 16, 2013 at 21:58 | answer | added | Ricardo Andrade | timeline score: 1 | |
| Dec 16, 2013 at 21:46 | comment | added | Allen Hatcher | The paper of LaBach mentioned in an earlier comment (now deleted?) uses a nonstandard topology on $Diff(D^n)$. The restriction map $Diff(D^n)\to Diff(int(D^n))$ is injective so can be viewed as an inclusion, and LaBach uses the subspace topology on $Diff(D^n)$ induced from the compact-open topology on $Diff(int(D^n))$. In this topology one can do a sort of "reverse Alexander trick" and push all the complications of a diffeomorphism of $D^n$ out to $\partial D^n$ and make them disappear. | |
| Dec 16, 2013 at 21:19 | comment | added | Igor Belegradek | Ricardo, before asking the question I looked at the Tom Goodwillie's answer, and thought the map $Diff N\to Emb(N, V)$ he considers is different from mine. His map is a restriction. Mine is precomposing with a given inclusion. Are you saying the maps are basically the same? | |
| Dec 16, 2013 at 21:17 | comment | added | Ricardo Andrade | As is essentially pointed out in this answer by Tom Goodwillie, the homotopy fibre of the map $\operatorname{Diff} N \to \operatorname{Emb}(N,V)$ is homotopy equivalent to the pseudo-isotopy space of $\partial N$, i.e. the space of diffeomorphisms of $\partial N \times I$ which fix $\partial N \times \{0\}$ pointwise. | |
| Dec 16, 2013 at 21:00 | comment | added | Allen Hatcher | $Diff(D^n)$ is homotopy equivalent to $O(n)\times Diff(D^n\ rel\ D^{n-1})$ where $D^{n-1}$ is a disk in $\partial D^n$. The factor $Diff(D^n\ rel\ D^{n-1})$ can be identified with the pseudoisotopy space $Diff(D^{n-1}\times I\ rel\ D^{n-1} \times 0)$, so it has a complicated homotopy type when $n$ is large enough. | |
| Dec 16, 2013 at 20:54 | comment | added | Igor Belegradek | @IanAgol: doesn't your map go the other way? I am looking at the precomposition $Diff(N)\to Emb(N,V)$ while you look at the restriction that goes the other way. | |
| Dec 16, 2013 at 20:14 | comment | added | Ian Agol | I think $Emb(N,V)$ is weakly homotopy equivalent to $Diff_c(V)$, the compactly supported diffeomorphisms of $V$ (certainly any isotopy can be realized as the restriction of a compactly supported isotopy; I think this works in families too). So I think you're looking to understand the exact sequence $Diff_{c,0}(V)=Diff_0(N,\partial N)\to Diff_0(N)\to Diff_0(\partial N)$. | |
| Dec 16, 2013 at 19:29 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Dec 16, 2013 at 19:26 | comment | added | Igor Belegradek | @Oscar Randal-Williams: thank you for the correction; I was confused here, and I shall edit the question. | |
| Dec 16, 2013 at 19:15 | comment | added | Oscar Randal-Williams | Then the paragraph concerning $\mathbb{R}^n$ and $D^n$ doesn't fit into the situation you are describing. If the boundary of $D^n$ is allowed to move, then $Diff(D^n) \simeq O(n)$ too. | |
| Dec 16, 2013 at 18:43 | comment | added | Igor Belegradek | @Oscar Randal-Williams: no, I do not assume that diffeomorphisms of $N$ are identity on the boundary. | |
| Dec 16, 2013 at 18:37 | comment | added | Oscar Randal-Williams | In the first paragraph, does $Diff(N)$ mean diffeomorphisms fixing the boundary pointwise or not? | |
| Dec 16, 2013 at 18:04 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Dec 16, 2013 at 17:41 | history | asked | Igor Belegradek | CC BY-SA 3.0 |