# 2 $MU(n)$ and $MU$

We define $MU(n)$ to be the Thom *space* of the tautological vector bundle of $BU(n)$.

We define $MU$ to be the Thom *spectrum* of the tautological virtual vector bundle of $BU$. Equivalently,

The direct sum map $BU \times BU \to BU$ induces a map $MU \wedge MU \to MU$, which turns $MU$ into a ring spectrum (and in fact an $\mathbb {E}_\infty$-ring spectrum).

Let $X$ be a space. Then $MU_*(X_+)$ admits the following description:

A class in $MU_d(X_+)$ is represented by a $d$-dimensional stably almost complex manifold $M$ and a map $f: M \to X$. Addition is given by disjoint union.

Two classes $f_1, f_2$ are equivalent if there is a cobordism $g: N \to X$ between them.

A map $S^d \to MU \wedge X_+$ factors through a map $S^d \to \Sigma ^{-2n} MU(n) \wedge X_+$ for some $n$, and is thus given a map of *spaces* $\phi : S^{d + 2n} \to MU(n) \wedge X_+$, increasing $n$ if necessary (since $\Sigma ^{2k} MU(n) \hookrightarrow MU(n + k)$). Recall that

where the point on the right is point at infinity. $MU(n)$ also contains the “zero section” isomorphic to $BU(n)$, and the normal bundle of the zero section is the tautological bundle of $BU(n)$. Generically, we can choose $\phi$ to intersect transversely in a way that $\phi ^{-1}(BU(n) \times X)$ is a codimension $2n$ submanifold of $S^{d + 2n}$, and the normal bundle, being pulled back from $BU(n)$, has an almost complex structure. This gives a stably almost complex manifold of dimension $d$ with a map to $X$. A homotopy of $\phi$ gives a cobordism between such maps.

So in particular, $MU_* = \pi _* MU$ is the complex cobordism group. Remarkably, these groups are entirely computable.

where $|m_i| = |b_i| = 2i$. The Hurewicz map $\pi _*(MU) \to H_*(MU)$ is injective but *not* surjective.