In a previous post, we studied the formal group law of ${\pi_* MU}$ in geometric terms: that is, using the interpretation of ${\pi_* MU}$ 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

$\displaystyle \log x = \sum_{i \geq 0 } \frac{[\mathbb{CP}^i]}{i+1} x^{i+1}.$

In particular, we saw that the complex projective spaces provided a set of independent generators for the rationalization ${\pi_* MU \otimes \mathbb{Q}}$: that is,

$\displaystyle \pi_* MU \otimes \mathbb{Q} \simeq \mathbb{Q}[\{ [\mathbb{CP}^i] \}_{i >0}].$

This is analogous to the theorem of Hirzebruch which calculates the oriented cobordism ring ${\pi_* MSO \otimes \mathbb{Q}}$, and could also have been established directly by arguing that ${\pi_* MU \otimes \mathbb{Q} \simeq H_*(MU; \mathbb{Q})}$. 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 ${\pi_* MU}$ (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 $MU$-cohomology. This is quite similar to the situation in ordinary cohomology, so this section will be brief.

Let ${E}$ 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 ${f: X \rightarrow Y}$, the stable normal bundle is the stable bundle ${f^* TY - TX + \mathbb{R}^N}$ for a high ${N}$.

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

$\displaystyle f_!: E^*(X) \rightarrow E^{* - \dim X + \dim Y}(Y),$

as follows. We have a factorization ${X \hookrightarrow V \rightarrow Y}$ for ${X \rightarrow V}$ an imbedding and ${V \rightarrow Y}$ a trivialized vector bundle of large dimension. A complex structure on the stable normal bundle of ${X}$ gives a complex structure on the stable normal bundle of ${X \hookrightarrow E}$. In particular, we can define a pushforward map

$\displaystyle E^*(X) \rightarrow E^{* + \dim_{\mathbb{R}} V - \dim X}(V, V - X) \rightarrow \widetilde{E}^{* + \dim_{\mathbb{R}} V - \dim X}(T(V)),$

where ${T(E)}$ is the Thom space of ${E}$. Now, use the suspension isomorphism (in fact, ${T(V)}$ is a suitable suspension of ${Y}$ as ${V}$ is trivial) to push this down to ${E^*(Y)}$.

Example: Let ${M}$ be a stably almost-complex manifold. Then there is a map ${M \rightarrow \ast}$, which is complex-oriented; the push-forward of the identity class gives an element of ${MU^{-\dim M}(\ast)}$, which is the class of ${M}$ in the cobordism ring.

Example: Let ${\mathcal{L}}$ be a complex line bundle on a manifold ${M}$ and let ${s}$ be a generic section (transverse to zero). Then the zero locus ${Z}$ of ${s}$ is such that the map ${i: Z \hookrightarrow X}$ admits a complex orientation. The pushforward ${i_!(1)}$ is the first Chern class of ${\mathcal{L}}$, 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 ${\pi_* MU}$ (together with the projective spaces). Consider the line bundle ${\mathcal{O}(1) \otimes \mathcal{O}(1)}$ on ${\mathbb{CP}^i \times \mathbb{CP}^j}$. We know that

$\displaystyle MU^*( \mathbb{CP}^i \times \mathbb{CP}^j) = \pi_* MU [t_1, t_2]/(t_1^{i+1}, t_2^{j+1}).$

Here ${t_1}$ is the first Chern class of ${\mathcal{O}(1)}$ on ${\mathbb{CP}^i}$, and similarly for ${t_2}$; that is, ${t_1}$ can really be thought of as living in ${MU^*(\mathbb{CP}^i)}$ and ${t_2}$ in ${MU^*(\mathbb{CP}^j)}$. It follows that

$\displaystyle c_1(\mathcal{O}(1) \otimes \mathcal{O}(1)) = F(t_1, t_2),$

where ${F}$ is the formal group law for ${\pi_* MU}$. In fact, we can be a little more precise: if ${F(x, y) = \sum a_{m,n} x^m y^n}$ is the formal group law, then

$\displaystyle c_1(\mathcal{O}(1) \otimes \mathcal{O}(1)) = \sum_{m\leq i, n \leq j} a_{m, n} t_1^m t_2^n.$

For ${f: M \hookrightarrow N}$ a complex-oriented inclusion, we will write ${[M \hookrightarrow N]}$ to denote the pushforward ${f_!(1) \in MU^*(N)}$. Now, if ${ \mathbb{CP}^{i-1} \hookrightarrow \mathbb{CP}^i}$ is the inclusion and ${: \mathbb{CP}^{j-1} \hookrightarrow \mathbb{CP}^j}$ is the other inclusion, we have

$\displaystyle t_1 = [\mathbb{CP}^{i-1} \hookrightarrow \mathbb{CP}^i] \in MU^*(\mathbb{CP}^i), \quad t_2 = [\mathbb{CP}^{j-1} \hookrightarrow \mathbb{CP}^j] \in MU^*(\mathbb{CP}^j).$

This follows because the ${\mathbb{CP}^{i-1}}$ is the vanishing locus of a good section of ${\mathcal{O}_{\mathbb{CP}^i}(1)}$. More generally, ${t_1^k = [\mathbb{CP}^{i-k} \hookrightarrow \mathbb{CP}^i]}$, and similarly for ${t_2}$. This is just the fact that multiplying the fundamental classes corresponds to intersecting cycles transversely.

Hence, we have

$\displaystyle c_1(\mathcal{O}(1) \otimes\mathcal{O}(1)) = \sum_{m \leq i, n \leq j} a_{m, n} [\mathbb{CP}^{i -m}\hookrightarrow \mathbb{CP}^i][\mathbb{CP}^{j-n} \hookrightarrow \mathbb{CP}^j].$

If we push forward along the map ${f: \mathbb{CP}^i \times \mathbb{CP}^j \rightarrow \ast}$, we get (using the properties of the Gysin map, which are analogous to those of ordinary cohomology)

$\displaystyle f_!(c_1(\mathcal{O}(1) \otimes \mathcal{O}(1)) = \sum_{m \leq i, n \leq j} a_{m, n} [\mathbb{CP}^{i-m}][\mathbb{CP}^{j-n}] \in \pi_* MU, \ \ \ \ \ (1)$

where ${[\mathbb{CP}^{i-m}]}$ denotes the cobordism class of ${\mathbb{CP}^{i-m}}$, or equivalently the pushforward of ${1}$ to a point from it. It follows that the classes ${f_!(c_1(\mathcal{O}(1) \otimes \mathcal{O}(1))}$ contain a fair bit of information about the generators ${a_{m, n}}$.

Anyway, the point is that ${c_1(\mathcal{O}(1) \otimes \mathcal{O}(1))}$ is the pushforward of the vanishing locus of a generic section. Suppose ${i \leq j}$. If we take the sections ${x_k, k \leq i}$ of ${\mathcal{O}_{\mathbb{CP}^i}(1)}$ and the sections ${y_k, k \leq i}$ of ${\mathcal{O}_{\mathbb{CP}^j}(1)}$, then we get a section ${x_1y_1 + \dots + x_i y_i}$ of ${\mathcal{O}(1) \otimes \mathcal{O}(1)}$. The zero section of this is the hypersurface

$\displaystyle H_{i,j} \subset \mathbb{CP}^i \times \mathbb{CP}^j$

cut out by the equation (in homogeneous coordinates) ${x_1 y_1 + \dots + x_i y_i = 0}$. These are the Milnor hypersurfaces.

By construction, we have

$\displaystyle c_1(\mathcal{O}(1) \otimes \mathcal{O}(1)) = [H_{i,j} \hookrightarrow \mathbb{CP}^i \times \mathbb{CP}^j].$

This implies that we have an identity in the complex cobordism ring:

$\displaystyle { [H_{i,j}] = \sum_{m \leq i, n \leq j} a_{m, n} [\mathbb{CP}^{i-m}][\mathbb{CP}^{j-n}]. } \ \ \ \ \ (2)$

In particular, modulo decomposables, we get that ${H_{i,j} \equiv a_{i,j} }$. This is valid for ${i, j > 1}$.

Corollary 2 The ${H_{i,j}}$ together with the ${\mathbb{CP}^i}$ generate the cobordism ring ${\pi_* MU}$.

Proof: We have, in fact, seen that the subring generated by the ${H_{i,j}}$ contains something congruent to each of the ${a_{i.j}}$ mod decomposables when ${i, j > 1}$. We need to handle the case when ${i}$ or ${j}$ is equal to ${1}$. But here we use the fact that

$\displaystyle \sum [\mathbb{CP}^n] u^n = \left( \sum_{r \geq 0} a_{r1} u^r \right)^{-1} .$

This is a general fact about log series: the log series of a formal group law given by ${f(x,y) = \sum a_{i,j} x^i y^j}$ satisfies

$\displaystyle \frac{d}{dx}\log x = \left( \frac{\partial}{\partial y} f(x, 0) \right)^{-1}.$

This equation shows that we can get the ${a_{r1}}$ mod decomposables with the projective spaces. Consequently, we can approximate any of the ${a_{i,j}}$ mod decomposables in the ring generated by the ${H_{i,j}}$ and the ${\mathbb{CP}^i}$, so this ring must be the whole thing. $\Box$