Pontryagin–Thom construction

Builds on Smooth manifold Submanifold Thom space

Definition

Let $M$ be a smooth manifold and let $N \subset M$ be a compact smooth submanifold without boundary, of codimension $k$ , with normal vector bundle $\nu$ . Choose an inner product on $\nu$ and a closed tubular neighbourhood $U$ of $N$ . Let $e \colon D(\nu) \to U$ be a diffeomorphism from the unit disc bundle that is the identity on the zero section. Let $\operatorname{Th}(\nu) = D(\nu) / S(\nu)$ be the Thom space of $\nu$ with base point $*$ . The Pontryagin–Thom collapse map $c \colon M \to \operatorname{Th}(\nu)$ is the following map: $c(x) = \begin{cases} [e^{-1}(x)] & \text{if } x \in U, \\ * & \text{if } x \notin U. \end{cases}$ Suppose that $\nu$ comes with a framing, that is, a trivialisation $\nu \cong N \times \R^k$ . The map $\operatorname{Th}(N \times \R^k) \to D^k / S^{k-1} \cong S^k$ forgets the point of $N$ . Its composite with $c$ is a map $M \to S^k$ .

Remarks

The map $c$ is continuous because the boundary of $U$ is the image of $S(\nu)$ , which goes to the base point; its homotopy class does not depend on the choices. Example: for $N$ a point of $M = S^n$ , the normal bundle is $\R^n$ with its standard framing, and $c \colon S^n \to D^n / S^{n-1} \cong S^n$ has degree $\pm 1$ . Non-example of the framed version: the circle $N = \R P^1$ in $M = \R P^2$ has the Möbius bundle as normal bundle, which has no framing. The collapse map still exists, with target $\operatorname{Th}(\nu) \cong \R P^2$ , but it does not give a map to $S^1$ .

For $M = S^{n+k}$ , Pontryagin's theorem says that the construction gives a bijection between framed cobordism classes of closed framed $n$ -dimensional submanifolds of $S^{n+k}$ and the homotopy group $\pi_{n+k}(S^k)$ . The inverse sends a smooth map, made transverse to a point, to the preimage of that point with the pulled back framing.

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Tag	`T9A1DZ4AQK`
URI	`m2://T9A1DZ4AQK`
Kind	definition
Writing	W0 stub
MSC	57R90, 55N22, 57R75
Defines	Pontryagin–Thom construction
Uses	Smooth manifold, Submanifold, Thom space
Changed	2 November 2026

Pontryagin–Thom construction

Definition

Remarks

Reading path

Further reading

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