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complex cobordism cohomology theory

Definition

Complex cobordism cohomology theory, denoted $MU$, is a generalized (Eilenberg-Steenrod) cohomology theory.

Its representing spectrum, also denoted $MU$ is the spectrum is in degree $2n$ given by the Thom space? of the vector bundle that is associated by the defining representation of the unitary group $U(n)$ on ${ℂ}^n$ to the universtal $U(n)$-principal bundle:

$$MU(2n)=\mathrm{Thom}\left(\mathrm{standard}\mathrm{associated}\mathrm{bundle}\mathrm{to}\mathrm{universal}\mathrm{bundle}\begin{array}{c}EU(n)\\ \downarrow \\ BU(n)\end{array}\right)$$M U(2n) = Thom \left( standard associated bundle to universal bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right)

The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:

$$MP={\vee }_{n\in \mathbb{Z}}{\Sigma }^{2n}MU$$M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U

There is a canonical orientation? on this obtained from the map

$$\omega :ℂP^{\mathrm{\infty }}\stackrel{\simeq }{\to }MU(1)MU(ℂP^{\mathrm{\infty }})$$\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty)

(???)

this is the universal even periodic cohomology theory with orientation

The cohomology ring $MP(*)$ is the Lazard ring which is the universal coefficient ring for formal group laws.

Related entries

for further context see the discussion at

• A Survey of Elliptic Cohomology - cohomology theories