We define to be the Thom space of the tautological vector bundle of .
We define to be the Thom spectrum of the tautological virtual vector bundle of . Equivalently,
The direct sum map induces a map , which turns into a ring spectrum (and in fact an -ring spectrum).
The crucial insight of Thom was that this spectrum is closely related to cobordism theory.
Let be a space. Then admits the following description:
A class in is represented by a -dimensional stably almost complex manifold and a map . Addition is given by disjoint union.
Two classes are equivalent if there is a cobordism between them.
[Proof idea] Given a class in
, I explain how to get such a map
. By the Whitney embedding theorem.
A map factors through a map for some , and is thus given a map of spaces , increasing if necessary (since ). Recall that
where the point on the right is point at infinity. also contains the “zero section” isomorphic to , and the normal bundle of the zero section is the tautological bundle of . Generically, we can choose to intersect transversely in a way that is a codimension submanifold of , and the normal bundle, being pulled back from , has an almost complex structure. This gives a stably almost complex manifold of dimension with a map to . A homotopy of gives a cobordism between such maps.
So in particular, is the complex cobordism group. Remarkably, these groups are entirely computable.
where . The Hurewicz map is injective but not surjective.
[Proof sketch] Since the tautological vector bundle of
is oriented, the first part follows from the Thom isomorphism theorem and the standard calculation of the homology of
. The homotopy groups follow from a more involved Adams spectral sequence calculation.