Complex cobordism

Definition

For $n \ge 0$, let $BU(n)$ be the classifying space of the unitary group $U(n)$. Let $\gamma_n \to BU(n)$ be the universal complex vector bundle of rank $n$, and let $MU(n)$ be its Thom space. The bundle $\gamma_n \oplus \C$ is classified by a map $BU(n) \to BU(n+1)$ covered by a bundle map $\gamma_n \oplus \C \to \gamma_{n+1}$, which induces a map $\Sigma^2 MU(n) \to MU(n+1)$ of Thom spaces. The complex cobordism spectrum $MU$ is the spectrum with $$MU_{2n} = MU(n), \qquad MU_{2n+1} = \Sigma MU(n),$$ whose structure maps are the identity $\Sigma MU_{2n} = MU_{2n+1}$ and the map $\Sigma MU_{2n+1} = \Sigma^2 MU(n) \to MU(n+1) = MU_{2n+2}$. Complex cobordism is the generalized cohomology theory represented by $MU$: for a pointed CW complex $X$ and $k \in \Z$, $$\widetilde{MU}^k(X) = \operatorname{colim}_n \, [\Sigma^n X, MU_{n+k}]_*.$$