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}. (more…)