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