# Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is an infinite-dimensional manifold; moreover, it is the total space of a smooth principal fiber bundle with structure group $\mathrm{Diff}(S)$ which has a base manifold $B(S,M)$ consisting of all submanifolds of $M$ of type $S$: $\require{AMScd}$ \begin{CD} \mathrm{Diff}(S) @>>> \mathrm{Emb}(S,M) \\ @. @VV{\pi}V \\ @. B(S,M) \end{CD}

Now, let's consider the diffeomorphism group $\mathrm{Diff}(M)$ of the ambient space $M$ rather than $S$. Then it seems that it has a natural action on $\mathrm{Emb}(S,M)$ (and even on $B(S,M)$) by $$\Big( \phi, (i:S\to M) \Big) \mapsto ( \phi\circ i :S\to M )$$ where $\phi\in \mathrm{Diff}(M)$.

Question: It would be interesting to study the properties of this $\mathrm{Diff}(M)$ action on $\mathrm{Emb}(S,M)$, like what are the orbits, the orbit space, etc. Moreover, it seems that this action preserves the fibers, so it should induce an action on the base $B(S,M)$. Is this true? Anyway, I fail to find an answer in literature. Does anyone know any reference?

• Yes, certainly it acts on $B(S,M)$: if $X \subset M$ is a submanifold diffeomorphic to $S$, then so is $\varphi(X) \subset M$. The action is transitive on connected components of $B(S,M)$ by an open-closed argument using the isotopy extension lemma. So what you are really describing is, if you fix an embedded submanifold $i: S \in B(S,M)$ (to pick a connected component), a fibration $\text{Diff}(M, S) \to \text{Diff}(M) \to B(S, M)_i$, where the fiber is the set of diffeomorphisms that fix $S$ as a set, ie $\varphi(S) = S$. – Mike Miller Aug 6 '18 at 21:06
• This is interesting and somewhat useful, as it related diffeomorphisms away from $S$ (and normal bundle data), diffeomorphisms of $M$, and a space of submanifolds of $M$. But you need to get control on 2 of the 3 to make much use out of it. – Mike Miller Aug 6 '18 at 21:08
• @MikeMiller Thank you very much, Mike. Would you mind mentioning some references on this issue, like why it is useful, how to get controls to make use of it? – Hang Aug 6 '18 at 21:16
• @Hang: references: a key tool is the isotopy extension theorem, see e.g. [Local Triviality of the Restriction Map for Embeddings, R. Palais Comm. math. Helv. (1960) 34, 305-312], see eudml.org/doc/139198. You may also find helpful the discussion in ￼ The non-finite homotopy type of some diffeomorphism groups, P.L.Antonelli, D.Burghelea, P.J.Kahn, Topology 11, 1972, 1-49], sciencedirect.com/science/article/pii/0040938372900213 – Igor Belegradek Aug 7 '18 at 1:24
• @Hang : There are indeed two fiber bundles: one in your question and one in Palais paper. I am not sure how this helps your objective of understanding $\mathrm{Diff}(M)$ action on the space of submanifolds and its orbit space. Of course it depends on what you want. If you just wish to say that isotopic submanifolds are in the same $\mathrm{Diff}(M)$-orbit, then this is true by isotopy extension. Simularly, every disk in $B(S, M)$ lifts to $\mathrm{Diff}(M)$. – Igor Belegradek Aug 7 '18 at 15:03