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 . (more…)