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 1 The 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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