The Hurewicz map and the logarithm of complex cobordism

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.

Climbing Mount Bourbaki