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This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make it a separate question, as it is also related to one of my previous questions, "Naïve"cobordism?.

The question I refer to is about the space of $m$-dimensional submanifolds of $\mathbb R^n$, and the above answer describes it as the disjoint sum of spaces of the form $\mathrm{Emb}(M,\mathbb R^n) /\mathrm{Diff}(M)$. In particular, each diffeomorphism type of $m$-manifolds produces a separate connected component of this space.

My question is whether any kind of compactification of this space is known which would tie together all these components.

Two things that come to mind in connection to this are

(a) the Deligne-Mumford compactification of the moduli spaces of curves via stable curves;

(b) Vassiliev invariants which are constructed via adding to the space of smooth 1-submanifolds of $\mathbb R^3$ new points corresponding to certain non-embedding immersions.

Is there actually a relationship between these two? Are there any higher dimensional analogs of these constructions known?

My personal motivation: a map from a (say, connected) space $X$ to the above disjoint sum should give a fibration over $X$ with fibre an $m$-manifold (plus some additional structure describing this fibre as a submanifold of $\mathbb R^n$); whereas a map to that purported compactification would instead give a family over $X$ which "mostly" consists of such manifolds but allows for "jumping" between different diffeomorphism types through (more or less) controllable "mild" singularities. This would give a picture which I was asking about in my above question about "naïve" cobordism.

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    $\begingroup$ I think there is a notion of constructible bundle of spaces over a stratified space that might answer your personal motivation, even if it doesn't result in compact moduli spaces (it seems like compactness is not what you care about anyway; it's not like you're trying to do intersection theory). This is mentioned in arxiv.org/abs/1502.01713, for example. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 7, 2015 at 8:46
  • $\begingroup$ @QiaochuYuan Thanks for the interesting link! You are right, "compactness" in the question is misleading. Rather I am thinking about things like models of the classifying spaces for cobordisms which presumably cannot be compact. Compactness here should be in a sense similar to that in which, say, $\mathbb CP^\infty$ or other Eilenberg-MacLane spaces are "compact", but I do not know how to define this kind of "compactness" precisely. The mother of all these "compact" things is the infinite-dimensional sphere (with the colimit topology of the equatorial embeddings of spheres into each other)... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 7, 2015 at 8:56

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I have a piece of juvenilia on this topic, considering the case of zero-dimensional submanifolds. See Section 9 of

O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds, IMRN 3 (2011) 572-608.

It concerns how close the relationship between

  1. the cobordism category having oriented 0-manifolds inside $M$ as objects, and oriented 1-dimensional cobordisms in $M \times [0,t]$ as morphisms, and
  2. the fundamental (topological) groupoid of McDuff's space of annihilating positive and negative particles in $M$,

is.

The main theorem in this direction is that while the categories are in no sense equivalent as topological categories (the circle as a cobordism $\emptyset \leadsto \emptyset$ is contractible as a loop in McDuff's space), they do nontheless have homotopy equivalent classifying spaces.

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