In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:

$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{framed}}.$$

Later on, in his 1954 paper, Thom generalises this with the now called Thom spaces, and shows that there is a similar correspondence for more general types of cobordism: manifolds with a $(B,f)$ structure on their normal bundle; for example, unoriented cobordism for $B = BO$, oriented cobordism for $B = BSO$, complex cobordism for $B = BU$, framed cobordism for $B = BI$ for the identity $I$ in $O$, etc. (Thom considers the cases $BO$ and $BSO$.)

The generalisation he arrived to, now called the Thom-Pontryagin construction, is the following:

$$\lim_{k \to \infty}\pi_{n+k}(TB_k) \cong \Omega_n^{(B,f)}, $$

where $TB$ is the Thom space of the universal bundle over $B$ given by the classifying map $B \to BO$; $TB$ is obtained by adding a point at infinity to each fiber and gluing all these added points to a single point in the total space of the bundle.

In fact, the result can be generalised further by considering cobordism as a homology theory, and one arrives at the following:

$$\Omega_n^{(B,f)}(X,Y) \cong \lim_{k \to \infty} \pi_{n+k}(X/Y \wedge TB_k),$$

where, if $Y$ is empty, $X/Y$ is the disjoint union of $X$ with a point (and $\wedge$ is the smash product). Here $\Omega_n(X,\emptyset)$ is to be understood as a relative cobordism over $X$. This clearly generalises the previous result by taking $X$ to be a point (and $Y$ empty).

Now, my question is, how do you understand the Thom-Pontryagin construction? I've seen a few mentions of a particularly visual way of understanding it, but without much to actually back this up (besides from, I remember, a few mentions of blobs of ink). The standard proofs (in Stong's *Notes on Cobordism Theory* or in Thom's original paper for example) are quite long and I have trouble keeping hold of my geometric intuition throughout.

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