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  "body": "\n## Definition\n\nFor $n \\ge 0$, let $BU(n)$ be the [[classifying space|WSZ40H5E4T]] of the unitary group $U(n)$.\nLet $\\gamma_n \\to BU(n)$ be the [[universal|1HDH716Z24]] complex vector bundle of rank $n$, and let $MU(n)$ be its [[Thom space|5MNYT3AFPH]].\nThe 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.\nThe *complex cobordism spectrum* $MU$ is the [[spectrum|1GHF4EKBXS]] with\n$$MU_{2n} = MU(n), \\qquad MU_{2n+1} = \\Sigma MU(n),$$\nwhose structure maps are the identity $\\Sigma MU_{2n} = MU_{2n+1}$ and the map $\\Sigma MU_{2n+1} = \\Sigma^2 MU(n) \\to MU(n+1) = MU_{2n+2}$.\n*Complex cobordism* is the [[generalized cohomology theory|H28PSZR7XG]] represented by $MU$: for a pointed CW complex $X$ and $k \\in \\Z$,\n$$\\widetilde{MU}^k(X) = \\operatorname{colim}_n \\, [\\Sigma^n X, MU_{n+k}]_*.$$\n\n## Remarks\n\n$MU$ is a [[Thom spectrum|6H8AGEN6NY]], and it is a commutative ring spectrum, with multiplication induced by the Whitney sum maps $BU(n) \\times BU(m) \\to BU(n+m)$.\nBy the Pontryagin–Thom construction, $\\pi_n(MU)$ is the [[cobordism|T8BVGPPSZP]] group of closed smooth $n$-manifolds with a complex structure on the stable normal bundle.\nMilnor and Novikov showed that $\\pi_*(MU) \\cong \\Z[x_1, x_2, \\ldots]$ with $\\deg x_i = 2i$; in particular $\\pi_n(MU) = 0$ for odd $n$.\n\nExample: $\\pi_2(MU) \\cong \\Z$, generated by the class of $\\C P^1$ with its complex structure.\n\nNon-example: $\\R P^2$ is a closed smooth surface, but it is not orientable, so it has no stably complex structure and gives no class in $\\pi_2(MU)$.\n",
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