In earlier posts, we analyzed the Hurewicz homomorphism

\displaystyle \pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq \mathbb{Z}[b_1, b_2, \dots ]

in purely algebraic terms: {\pi_* MU} was the Lazard ring, and the Hurewicz map was the map classifying the formal group law {\exp( \exp^{-1}(x) + \exp^{-1}(y))} for {\exp(x)} the “change of coordinates”

\displaystyle \exp(x) = x + b_1 x + b_2 x^2 + \dots.

We can also express the Hurewicz map in more geometric terms. The ring {\pi_* MU} is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring {H_*(MU; \mathbb{Z})} is isomorphic to the Pontryagin ring {H_*(BU; \mathbb{Z})} by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1 The Hurewicz map {\pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq H_*(BU; \mathbb{Z})} sends the cobordism class of a stably almost complex manifold {M} to the element {f_* [M]} where

\displaystyle f_* : M \rightarrow BU

classifies the stable normal bundle of {M} (together with its complex structure), and where {[M]} is the fundamental class of {M}.

In particular, we will be able to work out explicitly where a given complex manifold representing a cobordism class goes under this map. As an application, we’ll show using Lagrange inversion that the complex projective spaces determine the logarithm of the formal group law on {\pi_* MU}.

1. Proof of the theorem

This isn’t a deep theorem: in fact, we’ll just be chasing some diagrams and unwinding some definitions. First, let’s recall how the Thom-Pontryagin construction works. Let {M} be a stably almost-complex manifold. We can imbed {M \hookrightarrow S^N} for a high {N}, and in this case the normal bundle to {M} “is” the stable normal bundle for {N} large. In particular, the normal bundle {\nu \rightarrow M} of the imbedding {M \hookrightarrow S^N} is a complex bundle, at least if {N \equiv \dim M \mod 2}.

The Thom-Pontryagin collapse map

\displaystyle S^N \rightarrow T( \nu)

uses the fact that, by the tubular neighborhood theorem, the normal bundle {\nu} can be identified with a small neighborhood of {M} in {S^N}; the collapse map just collapses everything outside this neighborhood to the “point at infinity” in the Thom space.

Let {m = \dim_{\mathbb{C}} \nu}. Then, {T(\nu) } maps to the “universal” Thom space {T(\xi_m)} (of the universal bundle on {BU(m)}) in virtue of the classifying map

\displaystyle M \rightarrow BU(m)

classifying the complex structure on {\nu}. Composing, we get a map

\displaystyle S^N \rightarrow T(\nu) \rightarrow T(\xi_m) = \Sigma^{2m}MU(m),

which is precisely the element of {\pi_* MU} desired. So, we are interested in where the fundamental class of {S^N} goes in the homology of {MU}, or equivalently the homology of {BU}.

Now we have a commutative diagram

One checks that the fundamental class of {S^N} goes to the homology Thom class of {T(\nu)}; this corresponds under the Thom isomorphism to the fundamental class of {M}. The dual assertion is that the Thom cohomology class of {T(\nu)} pulls back to the fundamental cohomology class of {S^N}.

In particular, the image of {[S^N]} in {\widetilde{H}_*(T(\xi_m))} corresponds to the image of {[M]} in {H_{\dim M}(BU(m))}. But this is precisely the assertion of the theorem.

Note incidentally that the same type of argument can be used for the Hurewicz homomorphism in {MO} (for mod {2} homology).

2. Example: projective spaces

As an illustration, let’s work out where the complex projective spaces {\mathbb{CP}^n} live in {H_*(MU; \mathbb{Z})}.

The stable normal bundle of {\mathbb{CP}^n} is classified by a map

\displaystyle f: \mathbb{CP}^n \rightarrow BU,

and, by what we’ve seen, the Hurewicz image of the class of {\mathbb{CP}^n} in {\pi_{2n}(MU)} is the image {f( [\mathbb{CP}^n]) \in H_*(BU; \mathbb{Z}) \simeq H_*(MU; \mathbb{Z})}.

What is {f_*([M])} in terms of the basis for {H_*(BU)}? We can determine this by noting that BU is an H space, and that there is a map

\displaystyle g: \mathbb{CP}^n \rightarrow BU

classifying the tangent bundle, such that the product of {f} and {g} from {M} to BU is trivial. Let’s work out {g( [\mathbb{CP}^n])}.

Recall that {T_{\mathbb{CP}^n} = \hom(\mathrm{Taut}, \mathrm{Taut}^{\perp})} for {\mathrm{Taut}} the tautological line bundle, so that

\displaystyle T_{\mathbb{CP}^n}\oplus \mathbb{C} = \mathrm{Taut}^{\oplus (n+1)}.

In particular, if {\ell: \mathbb{CP}^n \rightarrow BU(1) \rightarrow BU} classifies the tautological line bundle, then

\displaystyle g = (n+1) \ell \in [\mathbb{CP}^n, BU],

where {[\mathbb{CP}^n, BU]} is a monoid. So, {g( [\mathbb{CP}^n])} is the image of {[\mathbb{CP}^n]} under the map

\displaystyle \mathbb{CP}^n \stackrel{\ell \times \dots \times \ell}{\rightarrow} BU^{n+1} \rightarrow BU.

Using the comultiplication formula in {H_*(BU; \mathbb{Z})}, we find

\displaystyle g_*([\mathbb{CP}^n]) = \sum_{i_1 + \dots + i_{n+1} = n} b_{i_1} \dots b_{i_{n+1}}.

Since {f +g = 0 \in [\mathbb{CP}^n, BU]}, we find that {f_*([\mathbb{CP}^n])} is the antipode of this. To compute this antipode, it suffices to compute the antipode {Sb_n} of {b_n}. The antipode has the property that

\displaystyle H \mapsto H \otimes H \stackrel{S \otimes 1}{\rightarrow} H \otimes H \rightarrow H

is just the unit followed by the counit. In particular, we find that, for {n \geq 1}

\displaystyle \sum_i S(b_i) b_{n-i} = 0 ,

in virtue of the comultiplication {b_n \mapsto \sum b_i \otimes b_{n-i}}. It follows that the series {\sum b_i x^i} and {\sum S(b_i) x^i} are multiplicative inverses to each other.

It follows that

\displaystyle f_*( [\mathbb{CP}^n]) = S( g_*( \mathbb{CP}^n)) = \sum_{i_1 + \dots + i_{n+1} = n} S(b_{i_1}) \dots S(b_{i_{n+1}}).

In particular, we find

\displaystyle f_*( [\mathbb{CP}^n]) = \left( \sum b_i x^i \right)^{-n-1}_n, \ \ \ \ \ (1)

where the subscript {n} indicates the degree {n} portion. This is the description of the Hurewicz image of {\mathbb{CP}^n}.

3. Lagrange inversion

The final goal of this post is to show that the equation (1) describing the Hurewicz images of the complex projective spaces in {H_*(MU; \mathbb{Z}) \simeq \mathbb{Z}[b_1, b_2, \dots ]} actually implies that the projective spaces determine the logarithm of the formal group law.

To see this, we will need a combinatorial formula for the inverse of a power series.

Proposition 2 (Lagrange inversion) Let {R } be a {\mathbb{Q}}-algebra, and let {\sum_{i \geq 0} c_i x^{i+1}} be a power series in {R[[x]]} with {c_0} invertible. Then the inverse power series is given by {\sum_{i \geq 0} d_i x^{i+1}} where

\displaystyle d_n = \frac{1}{n+1} \left( \sum c_i t^i \right)^{-n-1}_n,

where the subscript {n} denotes terms of degree {n}, as before.

Proof: Let {y = \sum c_i x^{i+1}}, so that {x = \sum_{i \geq 0} d_i y^{i+1}}. We will use the fact that the residue of a “1-form” is invariant under change-of-coordinates. Then we have

\displaystyle d_n = \frac{1}{n+1}\mathrm{Res}_{x=0}( \frac{dx}{y^n} ).

This was computed by expressing {x} as a power series in {y}. But we could equally well use the expression {y = \sum c_i x^{i+1}} here and, plugging that in for {y} and computing the residue via {x}, we get

\displaystyle d_n = \frac{1}{n+1} \left( \sum c_i t^i \right)^{-n-1}_n,

as desired. \Box

Now let’s combine this with what we know about formal group laws. Let {\exp(x) = \sum b_i x^{i+1} } be the exponential series over {H_*(MU; \mathbb{Z})}, as usual; then the formal group law of {H_*(MU; \mathbb{Z})} is

\displaystyle \exp(\exp^{-1}(x) + \exp^{-1}(y)).

This formal group law has a logarithm

\displaystyle \log x = \exp^{-1}(x) = \sum m_i x^{i+1},

where, by Lagrange inversion, we have

\displaystyle m_n = \frac{1}{n+1} \left( \sum b_i t^{i} \right)^{-n-1}_n.

Now if we combine this with (1), we have:

Corollary 3 (Miscenko)The log series for {\pi_* MU} is given by

\displaystyle \sum_{n \geq 0} \frac{1}{n+1} [\mathbb{CP}^n] x^{n+1}.

In particular, the complex projective spaces are generators mod torsion for the complex cobordism ring. This is analogous to the situation with oriented cobordism, where the even-dimensional complex projective spaces are generators.