Complex oriented cohomology theoriesMU(n)MU(n) and MUMU

2 MU(n)MU(n) and MUMU

Definition 2.1

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

We define MUMU to be the Thom spectrum of the tautological virtual vector bundle of BUBU. Equivalently,

MU=colimnΣ2nΣMU(n). MU = \operatorname*{colim}_n \Sigma ^{-2n} \Sigma ^\infty MU(n).

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

The crucial insight of Thom was that this spectrum is closely related to cobordism theory.
Theorem 2.2 (Pontryagin–Thom)

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

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

  2. Two classes f1,f2f_1, f_2 are equivalent if there is a cobordism g:NXg: N \to X between them.

[Proof idea] Given a class in MUd(X+)MU_d(X_+), I explain how to get such a map f:MXf: M \to X. By the Whitney embedding theorem.

A map SdMUX+S^d \to MU \wedge X_+ factors through a map SdΣ2nMU(n)X+S^d \to \Sigma ^{-2n} MU(n) \wedge X_+ for some nn, and is thus given a map of spaces ϕ:Sd+2nMU(n)X+\phi : S^{d + 2n} \to MU(n) \wedge X_+, increasing nn if necessary (since Σ2kMU(n)MU(n+k)\Sigma ^{2k} MU(n) \hookrightarrow MU(n + k)). Recall that

MU(n)X+=MU(n)×X/{}×X, MU(n) \wedge X_+ = MU(n) \times X / \{ *\} \times X,

where the point on the right is point at infinity. MU(n)MU(n) also contains the “zero section” isomorphic to BU(n)BU(n), and the normal bundle of the zero section is the tautological bundle of BU(n)BU(n). Generically, we can choose ϕ\phi to intersect transversely in a way that ϕ1(BU(n)×X)\phi ^{-1}(BU(n) \times X) is a codimension 2n2n submanifold of Sd+2nS^{d + 2n}, and the normal bundle, being pulled back from BU(n)BU(n), has an almost complex structure. This gives a stably almost complex manifold of dimension dd with a map to XX. A homotopy of ϕ\phi gives a cobordism between such maps.


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

Proposition 2.3
H(MU)=Z[b1,b2,b3,]π(MU)=Z[m1,m2,m3,] \begin{aligned} H_*(MU) & = \mathbb {Z}[b_1, b_2, b_3, \ldots ]\\ \pi _*(MU) & = \mathbb {Z}[m_1, m_2, m_3, \ldots ] \end{aligned}

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

[Proof sketch] Since the tautological vector bundle of BUBU is oriented, the first part follows from the Thom isomorphism theorem and the standard calculation of the homology of BUBU. The homotopy groups follow from a more involved Adams spectral sequence calculation.