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In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ on page 6 with the help of the

"standard $C^\infty$ topology on the space of d-dimensional smooth compact submanifolds of B that are properly embedded (meaning that the intersection of the submanifold with $\partial B$ is the boundary of the submanifold, and this is a transverse intersection)",

where $B$ is a closed ball centered in the origin.

I am searching for a reference or an explicit definition of the "standard topology" Hatcher is referring to.

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It's in Hirsch's "Differential Topology" textbook. Specifically, given a compact manifold $M$, the weak $C^k$-topology on the set of $C^k$-smooth embeddings $Emb(M,\mathbb R^n)$ is the one that requires uniform closeness of not only the maps themselves, but all derivatives up to order $k$. For $C^\infty$ mapping spaces you demand uniform agreement up to some arbitrary order $k$ -- it is not a normable space anymore so you think of the topology as being induced by a countable sequence of (semi)norms, one for each $k$.

That's the space of embeddings. So the space of submanifolds of $\mathbb R^n$ diffeomorphic to $M$ is the space

$$Emb(M,\mathbb R^n) / Diff(M) $$

where $Diff(M)$ is given the same "weak" topology as in Hirsch. $Diff(M)$ is the space of $C^k$-diffeomorphisms of $M$. Notice two embeddings $M \to \mathbb R^n$ have the same image if and only if they differ by a diffeomorphism of $M$.

Then the space of all $m$-dimensional manifolds in $\mathbb R^n$ is the disjoint union:

$$\sqcup_M Emb(M, \mathbb R^n) / Diff(M)$$

where you take the disjoint union over all diffeomorphism types of $m$-dimensional manifolds $M$.

It's a jazzed-up version of the Whitney embedding theorem that states $Emb(M,\mathbb R^n)$ is highly-connected for $n$ much larger than $m$.

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  • $\begingroup$ Does everything you say also work with manifolds with boundary? Is the space of submanifolds of a closed ball Hatcher talks about then induced by the subspace topology of the space you described? $\endgroup$
    – Madeleine
    Commented Jan 31, 2015 at 19:36
  • $\begingroup$ Yes, it also works although the argument is twice as thick in a sense. Usually you want relative embeddings $(M,\partial M) \to (D^n, \partial D^n)$. It's the same argument in spirit, just more longwinded. $\endgroup$
    – Ryan Budney
    Commented Jan 31, 2015 at 20:06
  • $\begingroup$ I thought that topologies induced by sequences of metrics were always metrizable. $\hspace{1.46 in}$ $\endgroup$
    – user5810
    Commented Jan 31, 2015 at 20:21
  • $\begingroup$ Sorry this might be offtopic wrt OP but this answer reminded me of something which I probably should better formulate as a separate question but maybe somebody can tell me here as well. By analogy with moduli spaces, does also in this setting exist a compactification of the above space whose points could be interpreted as manifolds with controllable singularities and which would tie together these connected components? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 1, 2015 at 16:08
  • $\begingroup$ One can topologize the collection of all topological submanifolds using the Vietoris topology or the Hausdorff metric. I believe that one can refine this topology for smooth manifolds to take into consideration the differential structure of the smooth manifolds. Is this Vietoris-like topology equivalent to the Emb/Diff topology? $\endgroup$
    – Joseph Van Name
    Commented Oct 14, 2022 at 0:30

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