---
tag: HM23BV561Z
kind: statement
role: theorem
name: Thom's cobordism theorem
msc: [57R75, 55N22]
uses: [T8BVGPPSZP, C02WQY819R, 5MNYT3AFPH, JQCK7A210W, 1HDH716Z24, P976WYEFMB]
about: [T8BVGPPSZP, C02WQY819R]
sources:
  - url: https://en.wikipedia.org/wiki/Cobordism
---

## Statement

Let $\Omega_n^O$ be the group of [[cobordism|T8BVGPPSZP]] classes of closed smooth $n$-manifolds.
Let $\Omega_*^O = \bigoplus_{n \ge 0} \Omega_n^O$ be the unoriented [[cobordism ring|C02WQY819R]], with product induced by the Cartesian product of manifolds.
For $k \ge 1$, let $MO(k)$ be the [[Thom space|5MNYT3AFPH]] of the [[universal bundle|1HDH716Z24]] $\gamma^k$ of rank $k$ over $BO(k)$.

1. For $k \ge n + 2$, there is an isomorphism $\Omega_n^O \cong \pi_{n+k}(MO(k))$ with the [[homotopy group|P976WYEFMB]] of $MO(k)$.
2. The ring $\Omega_*^O$ is a polynomial ring over $\Z/2\Z$ with one generator $x_i$ in each degree $i \ge 2$ that is not of the form $2^j - 1$:
$$\Omega_*^O \cong (\Z/2\Z)[x_2, x_4, x_5, x_6, x_8, x_9, \ldots].$$
3. Two closed smooth $n$-manifolds are cobordant if and only if all their [[Stiefel–Whitney numbers|JQCK7A210W]] agree.

## Remarks

The isomorphism in part 1 is the Pontryagin–Thom construction.
A closed $n$-manifold is embedded in $\R^{n+k}$, and a tubular neighbourhood of it is mapped to the Thom space of its normal bundle.
Collapsing the rest of the sphere $S^{n+k}$ gives an element of $\pi_{n+k}(MO(k))$, and transversality gives the inverse.
Parts 2 and 3 then follow from a computation of the homotopy of $MO(k)$ in the stable range by mod $2$ cohomology and Steenrod squares.

Example: in low degrees, $\Omega_1^O = 0$, $\Omega_2^O \cong \Z/2\Z$ generated by $\R P^2$, $\Omega_3^O = 0$ and $\Omega_4^O \cong (\Z/2\Z)^2$, spanned by the classes $x_2^2$ and $x_4$.
For even $i$, the generator $x_i$ can be taken to be $\R P^i$.

The result was proved by René Thom in 1954.

## Explanations

### L1 human

Deciding whether a manifold is the boundary of another manifold sounds like a hard geometric problem.
Thom's theorem turns it into a computation in homotopy theory, and then solves it completely.
In the end, whether a manifold bounds is decided by a finite list of numbers modulo $2$, its Stiefel–Whitney numbers.

### L2 human

The Pontryagin–Thom construction is the model for many later results that turn geometric questions about manifolds into stable homotopy groups of Thom spectra.
The same method computes the oriented cobordism ring through $MSO(k)$, and complex cobordism through $MU(k)$, whose homotopy is described by [[Quillen's theorem|4E518M24MA]].
Part 3 combines Pontryagin's observation that cobordant manifolds have the same Stiefel–Whitney numbers with Thom's converse.