A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)

Question

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the Thom space of the normal bundle $S^{m}=(\mathbb{R}^m)^{+} \to Th(N_j)$ (which depends on the choice of tubular neighborhood). The homotopy class of the collpase map is however well defined as a function of the isotopy class of the embedding (notice the homotopy type of the Thom space depends only on the isotopy class of the embedding). The "collapse map" is, in laconic terms, nothing more than a well defined function of sets:

$$\{{ \text{embeddings } M \hookrightarrow \mathbb{R}^m \}/\sim_{\text{isotopy}}} =: E \longrightarrow \coprod_{[j] \in E} [S^m,Th(N_j)]$$

This might be fine for the most part but it seems desirable (even from the POV of applications) to have a version of this construction which works for families.

Question: Is there a "natural" (in the sense that it does what you expect it to do for families of embeddings) map (more accurately a homotopy class of maps) of spaces:

$$Emb(M,\mathbb{R}^m) \to \coprod_{[j] \in \pi_0(Emb(M,\mathbb{R}^m))} Map_*(S^m,Th(N_j))$$

Which on connected components induces the collapse map construction above?

Answer 1

The right framework of definitions is as follows.

1. You have a space $E=\text{Emb}(M,\mathbb{R}^n)$ of smooth embeddings, topologised in a way that respects all derivatives. In more detail, we give $C^\infty(M)$ the smallest topology such that the inclusion in $C(M)$ is continuous, as is the map $C^\infty(M)\to C^\infty(M)$ given by any smooth vector field. We then topologise $E$ as a subspace of $C^\infty(M)^n$. One needs to check that for sufficiently small open sets $U\subseteq E$, there is a continuous choice of diffeomorphisms $r_{e,e'}\colon\mathbb{R}^n\to\mathbb{R}^n$ for $(e,e')\in U^2$ such that $r_{e,e'}\circ e'=e$.
2. Over $E\times M$ there is a vector bundle $\mu$, whose fibre at $(e,m)$ is $\mathbb{R}^n\ominus e_*(T_mM)$. Thus, $\mu|_{\{e\}\times M}$ can be identified with the normal bundle $\nu_e$. One needs to check that for sufficiently small open sets $U\subseteq E$, the restriction $\mu|_{U\times M}$ is isomorphic to a bundle pulled back from $M$, so $\nu_e$ is essentially independent of $e$ for $e\in U$.
3. There is also a space $F$ of thickened embeddings. A point of $F$ is a pair $(e,f)$ where $e\in E$, and $f$ is an embedding of the total space of $\nu_e$ in $\mathbb{R}^n$, with $f(m,u)\simeq e(m)+u$ to first order in $\|u\|$. There is an evident way to topologise $F$ so that it has the same kind of $E$-local triviality as $\mu$.
4. One then needs to check that the projection $q\colon F\to E$ is a weak equivalence (or even a homotopy equivalence). This can be done by elaborating the standard proof of uniqueness of tubular neighbourhoods, combined with some paracompactness technology if you want an actual homotopy equivalence.
5. There is an evident fibrewise Pontrjagin-Thom construction $p\colon F_+\wedge S^n\to (F\times M)^{q^*\mu}$, and a projection $q'\colon (F\times M)^{q^*\mu}\to (E\times M)^\mu$. If we regard the homotopy category as a category of fractions in which weak equivalences are inverted, then we now have a morphism $p'=q'\circ p\circ(q\wedge 1)^{-1}\colon E_+\wedge S^n\to (E\times M)^\mu$. This is the most natural incarnation of the Pontrjagin-Thom construction.

Answer 2

The situation is somewhat easier to describe if one replaces the embeddings of the closed codimension $(m-n)$manifold $M$ with the embeddings of the total space of a disk bundle of a rank $(m-n)$-vector bundle over $M$.

If $\xi$ is a smooth vector bundle over $M$ of rank $(m-n)$, with disk bundle $D(\xi)$, then there is a restriction map of embedding spaces $$ E(D(\xi),\Bbb R^m) \to E(M,\Bbb R^m)\, , $$ There is an evident Pontryagin-Thom map $$ E(D(\xi),\Bbb R^m) \to \Omega^mM^\xi $$ where the target it the $m$-fold loop space of the Thom space of $\xi$. The map is given by sending an embedding to the map $S^m \to M^\xi$ given by collapsing the complement of the image of the embedding to a point.

The above restriction map sits in a homotopy fiber sequence, $$ E(D(\xi),\Bbb R^m) \to E(M,\Bbb R^m) \to \text{maps}(M,BO_{m-n}) $$ where the base space is the space of maps from $M$ to the Grasmannian of $(m-n)$-planes in $\Bbb R^\infty$. The displayed fiber is the one taken at the basepoint defined by $\xi$.

There is another homotopy fiber sequence $$ \Omega^m M^\xi \to D_m(M) \to \text{maps}(M,BO_{m-n}) $$ where $M^\xi$ is the Thom space of $\xi$ and $D_m(M)$ is the space consisting of pairs $(\xi,g)$ in which $\xi$ is as above and $g: S^m \to M^\xi$ is a based map.

The first fiber sequence maps to the second one, after thickening up $E(M,\Bbb R^m)$ in the way that Neil describes.