In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there.
He starts with a map $f \colon R^{n+k}\to R^n_+$ (constant outside a compact) coming from a map $f \colon S^{n+k}\to S^n$ where we identify the spheres with the one-point compactification of the respective real spaces (denoted with $R^n_+$), and he builds a fattened manifold $M^k\times E^n \xrightarrow{embedded} R^{n+k}$ using such map $f$. Then he introduce this map $\theta$, homotopic to the identity of $R^n_+$ which spreads $E^n$ on all $R^n$ and everything else to $\infty$. and now it comes the part which is confusing me:
Then the composition $\theta\circ f\simeq f$ is the same as the map taking $N\cong M^k\times E^n \to E^n$ by the projection followed by a diffeomorphism $E^n\to R^n$ and taking everything else to $\infty$. Therefore every fattened $k-$manifold $g\colon M^k\times E^n \to R^{n+k}$ gives rise to a map $\phi_g \colon R^{n+k} \to R^n_+$ of this form, and every map $R^{n+k} \to R^n_+$ as above, is homotopic to a map arising in this way.
I don't get how a fattened manifold induces the map $\phi_g$, because the only idea which came to my mind is to include the fat. manifold in $R^{n+k}$, project to the second factor $E^n$, and then use the map $\theta$ (as suggested by the picture above?) but in this way the induced map is always null-homotopic, because $E^n \neq S^n$ and therefore by composing with a stereographic projection we can "shrink" it to a point.
So how does this "assignment" works precisely?
I think, "fattern manifold" is $k$-submanifold $M\subset\mathbb R^{n+k}$ with trivialized normal bundle, so you can identify neighborhood of $M$ with $M\times E^n$, $g$ is the inclusion $M\times E^n\subset \mathbb R^{n+k}$. And the map $\theta_g$ is constructed as follows: you should project $M\times E^n$ on the $E^n$, then cover $S^n$ without $\infty$ by this $E^n$, and send all the $\mathbb R^{n+k}\setminus (M\times E^n)$ to $\infty$.
If you continue $\theta_g$ on the $S^{n+k}=\mathbb R^{n+k}\cup\{\infty\}$ (let $\infty\mapsto\infty$), this mapping is not always null-homotopic. For example, if $n=2$, $k=1$ and $M$ is $S^1$ with standart embedding into $\mathbb R^3$, the map $\theta_g:S^3\to S^2$ will be homotopic to Hopf fibration (you can prove is as a beautiful geometric-topology exercise).
EDIT: But fixed trivialization of normal bundle (or neighborhood) of $S^1$ with $S^1\times \mathbb R^2$ must be not "trivial", it must be twisted so that any two constant sections form the Hopf link.