In a previous post, we studied the formal group law of in geometric terms: that is, using the interpretation of as the cobordism ring of stably almost-complex manifolds. We found that the logarithm for this formal group law was given by the power series
In particular, we saw that the complex projective spaces provided a set of independent generators for the rationalization : that is,
This is analogous to the theorem of Hirzebruch which calculates the oriented cobordism ring , and could also have been established directly by arguing that . The structure of the latter ring can be worked out directly, and in fact was.
We might be interested, though, in a set of honest generators for (not generators mod torsion). Such a set is provided by the Milnor hypersurfaces which I would like to discuss in this post.
1. The pushforward maps
To start with, it will be convenient to have the language of Gysin (pushforward) maps in -cohomology. This is quite similar to the situation in ordinary cohomology, so this section will be brief.
Let be a complex-oriented cohomology theory. Then one has a theory of Gysin pushforward maps analogous to that in ordinary homology, in certain cases.
Definition 1 Given a map of manifolds , the stable normal bundle is the stable bundle for a high .
This is just a generalization of the notion of a stable normal bundle for an imbedding. Given a complex structure on the stable normal bundle, we can define a push-forward map
as follows. We have a factorization for an imbedding and a trivialized vector bundle of large dimension. A complex structure on the stable normal bundle of gives a complex structure on the stable normal bundle of . In particular, we can define a pushforward map
where is the Thom space of . Now, use the suspension isomorphism (in fact, is a suitable suspension of as is trivial) to push this down to .
Example: Let be a stably almost-complex manifold. Then there is a map , which is complex-oriented; the push-forward of the identity class gives an element of , which is the class of in the cobordism ring.
Example: Let be a complex line bundle on a manifold and let be a generic section (transverse to zero). Then the zero locus of is such that the map admits a complex orientation. The pushforward is the first Chern class of , as in ordinary cohomology.
2. The Milnor hypersurfaces
The next goal is to describe the Milnor hypersurfaces, which provide a system of generators for the cobordism ring (together with the projective spaces). Consider the line bundle on . We know that
Here is the first Chern class of on , and similarly for ; that is, can really be thought of as living in and in . It follows that
where is the formal group law for . In fact, we can be a little more precise: if is the formal group law, then
For a complex-oriented inclusion, we will write to denote the pushforward . Now, if is the inclusion and is the other inclusion, we have
This follows because the is the vanishing locus of a good section of . More generally, , and similarly for . This is just the fact that multiplying the fundamental classes corresponds to intersecting cycles transversely.
Hence, we have
If we push forward along the map , we get (using the properties of the Gysin map, which are analogous to those of ordinary cohomology)
where denotes the cobordism class of , or equivalently the pushforward of to a point from it. It follows that the classes contain a fair bit of information about the generators .
Anyway, the point is that is the pushforward of the vanishing locus of a generic section. Suppose . If we take the sections of and the sections of , then we get a section of . The zero section of this is the hypersurface
cut out by the equation (in homogeneous coordinates) . These are the Milnor hypersurfaces.
By construction, we have
This implies that we have an identity in the complex cobordism ring:
In particular, modulo decomposables, we get that . This is valid for .
Corollary 2 The together with the generate the cobordism ring .
Proof: We have, in fact, seen that the subring generated by the contains something congruent to each of the mod decomposables when . We need to handle the case when or is equal to . But here we use the fact that
This is a general fact about log series: the log series of a formal group law given by satisfies
This equation shows that we can get the mod decomposables with the projective spaces. Consequently, we can approximate any of the mod decomposables in the ring generated by the and the , so this ring must be the whole thing.