# Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the $C^\infty$-topology. Furthermore, if $M$ is compact, the quotient map $Emb^{C^\infty}(M,N) \to Emb^{C^\infty}(M,N)/Diff(M)$ is a principal bundle map.

My question is what is known about this in the setting of topological manifolds. In particular, if $M$ and $N$ are topological manifolds, it is easy to prove that the homeomorphisms $Homeo(M)$ act continuously on the topological embeddings $Emb(M,N)$ (a topological embedding is a continuous map that is a homeomorphism onto its image) if both have the compact-open topology. Thus there is a similar quotient map $Emb(M,N) \to Emb(M,N)/Homeo(M)$.

If $M$ is compact, is the quotient map $Emb(M,N) \to Emb(M,N)/Homeo(M)$ a principal bundle map?

Given how the proof goes in the smooth case, one might expect that it is in fact badly behaved, but sometimes the topological setting is nicer than expected. I would be interested in a reference to either a positive or a negative answer, but intermediate answers, like it being a Serre fibration, are also fine.

• Let me point out that a sequence of knotted smooth embeddings of a circle in 3-space can converge in the compact-open topology to an unknotted embedding. This should lead to a counterexample. – Tom Goodwillie Jan 2 '15 at 22:57
• (Are you sure you don't want to restrict to tame embeddings?) If Emb(M,N)→Emb(M,N)/Homeo(M) is a principal bundle, then Emb(M,N) x_Homeo(M) M → Emb(M,N)/Homeo(M) is a fiber bundle with fiber M. Maybe it's possible to show that the latter is not a fiber bundle, and thus solve the problem to the negative? – André Henriques Jan 16 '15 at 19:07
• I'm happy to restrict to tame, locally tame or locally flat embeddings if that allows us to say anything, but it isn't clear to me that makes the problem easier. – skupers Jan 16 '15 at 20:27