Thom's cobordism theorem

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.

Going deeper

For a student

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. Part 3 combines Pontryagin's observation that cobordant manifolds have the same Stiefel–Whitney numbers with Thom's converse.