In earlier posts, we analyzed the Hurewicz homomorphism

in purely algebraic terms: was the Lazard ring, and the Hurewicz map was the map classifying the formal group law for the “change of coordinates”

We can also express the Hurewicz map in more geometric terms. The ring is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring is isomorphic to the Pontryagin ring by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1The Hurewicz map sends the cobordism class of a stably almost complex manifold to the element where

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

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 .

**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 be a stably almost-complex manifold. We can imbed for a high , and in this case the normal bundle to “is” the stable normal bundle for large. In particular, the normal bundle of the imbedding is a complex bundle, at least if .

The **Thom-Pontryagin collapse map**

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

Let . Then, maps to the “universal” Thom space (of the universal bundle on ) in virtue of the classifying map

classifying the complex structure on . Composing, we get a map

which is precisely the element of desired. So, we are interested in where the fundamental class of goes in the homology of , or equivalently the homology of .

Now we have a commutative diagram

One checks that the fundamental class of goes to the homology Thom class of ; this corresponds under the Thom isomorphism to the fundamental class of . The dual assertion is that the Thom cohomology class of pulls back to the fundamental cohomology class of .

In particular, the image of in corresponds to the image of in . 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 (for mod homology).

**2. Example: projective spaces**

As an illustration, let’s work out where the complex projective spaces live in .

The stable normal bundle of is classified by a map

and, by what we’ve seen, the Hurewicz image of the class of in is the image .

What is in terms of the basis for ? We can determine this by noting that is an H space, and that there is a map

classifying the tangent bundle, such that the product of and from to is trivial. Let’s work out .

Recall that for the tautological line bundle, so that

In particular, if classifies the tautological line bundle, then

where is a monoid. So, is the image of under the map

Using the comultiplication formula in , we find

Since , we find that is the antipode of this. To compute this antipode, it suffices to compute the antipode of . The antipode has the property that

is just the unit followed by the counit. In particular, we find that, for

in virtue of the comultiplication . It follows that the series and are multiplicative inverses to each other.

It follows that

In particular, we find

where the subscript indicates the degree portion. This is the description of the Hurewicz image of .

**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 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 be a -algebra, and let be a power series in with invertible. Then the inverse power series is given by where

where the subscript denotes terms of degree , as before.

*Proof:* Let , so that . We will use the fact that the residue of a “1-form” is invariant under change-of-coordinates. Then we have

This was computed by expressing as a power series in . But we could equally well use the expression here and, plugging that in for and computing the residue via , we get

as desired.

Now let’s combine this with what we know about formal group laws. Let be the exponential series over , as usual; then the formal group law of is

This formal group law has a **logarithm**

where, by Lagrange inversion, we have

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

Corollary 3 (Miscenko)The log series for is given by

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.

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