Let $M$ and $N$ be smooth manifolds with $\dim M<\dim N$.
The spaces $\mathrm{Emb}(M,N)\subset\mathrm{Imm}(M,N)$ of smooth embeddings and immersions $f:M\to N$, respectively, are infinite dimensional Frechet manifolds. They are open subsets of the space $C^{\infty}(M,N)$ of smooth maps $M\to N$, or equivalently, of the space $\Gamma(M,M\times N)$ of smooth sections of the trivial bundle $M\times N\to M$.
The diffeomorphism group $\mathrm{Diff}(M)$ acts naturally on these spaces by precomposition.
My question is what structure the quotient spaces $\mathrm{Emb}(M,N)/\mathrm{Diff}(M)$ and $\mathrm{Imm}(M,N)/\mathrm{Diff}(M)$ have. These spaces seem to arise naturally in geometric problems, where only the image $f(M)\subset N$ of a map $f:M\to N$ is of concern, not the actual map itself.
1) Do $\mathrm{Emb}(M,N)/\mathrm{Diff}(M)$ and $\mathrm{Imm}(M,N)/\mathrm{Diff}(M)$ have a smooth manifold structure?
2) What are their topological properties, e.g. homology/homotopy? Can we compute these via some kind of "Morse" functions?
Best Answer
1) See this and its references. The issue in the case of immersions is when an immersion $f: M \to N$ factors through a covering map $g: M \to M'$.
2) For embeddings, this is a ridiculously hard question, analagous to understanding the homology and homotopy of the diffeomorphism group, which is the subject of much (ongoing) research. For instance, the examples of embedded 2-spheres inside $S^3$ are what allowed Hatcher to calculate the homotopy type of $\text{Diff}(S^1 \times S^2)$, and understanding those is essentially equivalent to his very difficult result that $\text{Diff}^+(D^3, \partial D^3)$ is contractible. Similarly, once he has the homotopy type of the space of Haken manifold here one may bootstrap that into an understanding of the topology of diffeomorphism groups.
For immersions this is answered by Hirsch-Smale theory.