The geometry of Chern numbers

Abstract

In this note we give an elementary proof of Milnor’s theorem that Chern numbers determine complex bordism. The proof is based on the idea of the authors’ previous paper [1] in which Thom’s theorem on unoriented bordism was given a similar proof. The reader is referred to this paper for an easier, more special argument illustrating our idea.

The theorem is proved by induction on dimension. One uses the theorem in dimensions $\lt n$ to show (1.4) that if $M^n$ has all Chern numbers zero, and $t_M\colon M\to \mathbf{BU}$ defines the weakly-complex structure on $M$, then $\{M,t_M\} = \{N,c\}$ in $\Omega_n^U(\mathbf{BU})$ where $c$ is a contant map. For each prime $p\ge 2$, this leads (§2) to a relation in $\Omega_\ast^U(\mathbf{BZ}/p)$, whence (§3) $p$ divides $\{M\}$ in $\Omega_n^U$. It then follows that $\{M\} = 0$, if we use the fact, from homotopy theory, that $\Omega_n^U$ is finitely generated for each $n$.