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…)