MU and Complex Orientations (Lecture 6)

February 4, 2010
In the last lecture, we defined spectra MU(n) = Σ∞−2n BU (n)/BU (n − 1) which form a direct system
MU(0) → MU(1) → MU(2) → · · ·The (homotopy) colimit of this sequence is called the complex bordism spectrum and is denoted by MU.
Example 1. The spectrum MU(0) is equivalent to the sphere spectrum.
Example 2. The spectrum MU(1) is the desuspension Σ∞−2 CP∞ of CP∞ = BU(1).
Remark 3. In terms of the above identifications, the inclusion MU(0) → MU(1) is given by
MU(0) ' S ' Σ∞−2 S 2 → Σ∞−2 CP∞ = MU(1).
Remark 4. The direct sum of complex vector bundles is classified by a multiplication ma,b : BU (a) ×BU (b) → BU (a + b). Passing to Thom spectra, we get a multiplication MU(a) ⊗ MU(b) → MU(a + b). We
note that the inclusion MU(n) → MU(n + 1) can be identified with the map
MU(n) ' S ⊗ MU(n) = MU(0) ⊗ MU(n) → MU(1) ⊗ MU(n) → MU(n + 1).
Remark 5. Taking the limit in a and b, we get a multiplication MU ⊗ MU → MU. That is, MU has thestructure of a ring spectrum. In fact, this multiplication is commutative and associative up to homotopy. It
has a unit, given by the map S ' MU(0) → MU.
In fact, the situation is much better: the multiplication on MU is commutative and associative up to all
higher homotopies. That is, MU has the structure of an E∞ -ring spectrum.
Let E be a complex-oriented cohomology theory. In the last lecture, we saw that every complex vector
bundle is E-orientable. In fact, for each integer n we can write down a canonical orientation of the universal
bundle ζn of rank n on the classifying space BU (n): it is classified by a map φn : MU(n) → E. These maps
φn are uniquely characterized by the following requirements:(1) The map φ1 : MU(1) → E is given by the complex orientation of E (note that we can identify the sete 2 (CP∞ ).
of homotopy classes of maps [MU(1), E] with E(2) The maps φn are multiplicative in the following sense: for every pair of integers m and n, the diagram
MU(m) ⊗ MU(n)φm ⊗φnE⊗Ecommutes up to homotopy.

1/ MU(m + n)
φm+n


/E

To prove assertion (2), we recall that E ∗ (MU(m + n)) can be identified (up to a shift in grading)with the ideal in E ∗ (BU(m + n)) ' (π∗ E)[[c1 , . . . , cm+n ]] generated by the Chern class cm+n . Similarly,E ∗ (MU(m)⊗MU(n)) can be identified with the ideal in E ∗ (BU(m)×BU(n)) ' (π∗ E)[[c1 , . . . , cm , c01 , . . . , c0n ]]generated by the product cm c0n . The commutativity of the diagram now follows from the equation cm+n (ζ ⊕ζ 0 ) = cm (ζ)cn (ζ 0 ), where ζ and ζ 0 are vector bundles of rank m and n.
We claim that the composite map
φn+1

MU(n) → MU(n + 1) → Ecoincides with φn . Using (2) and Remark 4, we deduce that this composite map is given by
φn φ1

MU(n) ' MU(n) ⊗ MU(0) → MU(n) ⊗ MU(1) → E.
We are therefore reduced to proving that φ1 | MU(0) coincides with φ0 (which is the unit map S → E).
According to Remark 3, this map is given by the class in π0 E given by restricting our complex orientatione 2 (CP∞ ) to E
e 2 (S 2 ) ' π0 E, which we have assumed to be the unit in E.
t∈E
The mapping spectrum E MU can be obtained as a homotopy limit of mapping spectra E MU(n) . We
therefore have a Milnor long exact sequencelim1 {E −1 (MU(n))} → E 0 (MU) → lim E 0 (MU(n)) → lim1 {E 0 (MU(n))}.
←−←−←−The outer groups vanish, since each of the restriction maps E ∗ (MU(n + 1)) → E ∗ (MU(n)) is surjective (itcorresponds, under our choice of Thom isomorphisms, to the restriction map E ∗ (BU (n + 1)) → E ∗ (BU (n)),
which is obtained by killing cn+1 in the power series ring E ∗ (BU (n + 1)) ' (π∗ E)[[c1 , . . . , cn+1 ]]). It follows
that the maps φn : MU(n) → E can be uniquely amalgamated to give a map φ : MU → E.
Proposition 6. The map φ is a map of ring spectra.
Proof. We must show that the diagram
MU ⊗ MU

φ⊗φ/ MUφE⊗E
/Ecommutes. Repeating the above argument, we conclude that E 0 (MU ⊗ MU) can be obtained as the inverse
limit of the cohomology groups E 0 (MU(a) ⊗ MU(b)). The desired result now follows from the commutativityof each of the squares/ MU(a + b)
MU(a) ⊗ MU(b)E⊗E/E(see Remark 4).
2g (CP∞ ). By construction, theThe inclusion Σ∞−2 CP∞ ' MU(1) → MU determines a class t ∈ MU
ring spectrum map φ : MU → E carries t to our chosen complex orientation of E.
2g (CP∞ ) is a complex orientation of MU. To see this, we note that theRemark 7. The class t ∈ MU
2g (S 2 ) ' π0 MU is given by the map S ' MU(0) → MU(1) → MU, which is the unit
restriction of t to MU
for the ring spectrum MU.
We can summarize our discussion as follows:22g (CP∞ ) be the complex orientation
Theorem 8. Let E be a commutative ring spectrum, and let t ∈ MU
described above. The construction (φ : MU → E) 7→ φ(t) determines a bijection between complex orientationsof E and ring spectrum maps MU → E.
In other words, the complex bordism spectrum MU is the universal complex-oriented cohomology theory.
Proof. The above analysis shows that given any complex orientation of E, we can construct a ring spectrum
map φ : MU → E which carries our complex orientation to the specified complex orientation of E. It remains
to prove injectivity. Let φ, φ0 : MU → E be two ring spectrum maps which determine the same complex
orientation of E; we wish to prove that φ and φ0 are homotopic. The condition that φ and φ0 determine thesame complex orientation tells us that φ| MU(1) ' φ0 | MU(1). Since E is complex-orientable, the precedingargument shows that E 0 (MU) ' lim E 0 (MU(n)). It will therefore suffice to show that φ and φ0 have the←−same restriction to MU(n) for every integer n. Since φ is a ring map, the composition
φMU(1)⊗n → MU(n) → E
is a product of n copies of φ| MU(1) in E 0 (MU(1)), and therefore coincides with the compositionφ0MU(1)⊗n → MU(n) → E.
It will therefore suffice to show that the restriction maps E 0 (MU(n)) → E 0 (MU(1)⊗n ). Using our Thomisomorphisms (provided by any complex orientation of E), this is equivalent to the injectivity of the map
E 0 (BU(n)) → E 0 (BU(1)n ), which is the “spliting principle” we discussed earlier.
Theorem 8 suggests that if we are interested in complex-oriented cohomology theories and the associatedformal group laws, then we should focus our attention on the complex bordism spectrum MU. The universal
complex orientation determines a (graded) formal group law f (x, y) ∈ (π∗ MU)[[x, y]]. As we have seen, thisformal group law is given by a map of graded rings L → π∗ MU.
Our goal next week will be to prove the following theorem:
Theorem 9 (Quillen). The map L → π∗ MU is an isomorphism. (In particular, the spectrum MU has
homotopy groups only in even degrees.)3