Introduction (Lecture 1)

January 22, 2010
A major goal of algebraic topology is to study topological spaces by means of algebraic invariants (suchas homology or cohomology). There is a balance to be struck here: we would like our invariants to be
simple enough to be tractable and computable, but rich enough to convey interesting information about
topology. In this course, we are going to study one example where both of these demands can be satisfied.
This is the so-called “chromatic” picture of stable homotopy theory, and it begins with Quillen’s work onthe relationship between cohomology theories and formal groups.
Let E be a multiplicative cohomology theory. For any topological space X, one can attempt to computethe E-cohomology groups E ∗ (X) by means of the Atiyah-Hirzebruch spectral sequence
Hp (X; E q (∗)) ⇒ E p+q (X).
If X is the infinite dimensional projective space CP∞ , then its ordinary cohomology groups are given
by H∗ (X; Z) ' Z[t], where t ∈ H2 (X; Z) is a generator. We say that E is complex-orientable if theAtiyah-Hirzebruch spectral sequence degenerates at the second page. In this case, we get an isomorphism
E ∗ (CP∞ ) ' E ∗ (∗)[[t]] for some generator t ∈ E 2 (∗).
In ordinary cohomology, we can define t ∈ H2 (CP∞ ; Z) to be the first Chern class c1 (O(1)), where O(1)denotes the universal line bundle on CP∞ . Conversely, if we are given t then we can define the first Chernclass in general, using the fact that CP∞ is a classifying space for complex line bundles. Namely, if L is anycomplex line bundle on a (nice) space X, then there exists a continuous map f : X → CP∞ (well-definedup to homotopy) and an isomorphism L ' f ∗ O(1). We can then define c1 (L) = f ∗ t ∈ H2 (X; Z).
If E is a complex-orientable cohomology theory, then the isomorphism E ∗ (CP∞ ) ' E ∗ (∗)[[t]] permits usto define a Chern class which takes values in E-cohomology. Namely, if L is a line bundle on a space X and∗2f : X → CP∞ is defined as above, then we can define cE1 (L) = f t ∈ E (X).
Warning 1. The definition of the Chern class cE1 (L) depends not only on the cohomology theory E,but also on the choice of isomorphism E ∗ (CP∞ ) ' E ∗ (∗)[[t]] (that is, on the choice of t). A complexorientable cohomology theory E together with a choice of generator t ∈ E 2 (CP∞ ) is called a complex-oriented
cohomology theory.
We now ask: how well-behaved is this theory of E-valued Chern classes? For example, ordinary Chernclasses satisfy a multiplicativity formula
c1 (L ⊗ L0 ) ' c1 (L) + c1 (L0 ).
Does the analogous formula hold in E-cohomology? Generally, the answer is no. However, we can say that00EE
there is always some formula which allows us to express cE1 (L ⊗ L ) in terms of c1 (L) and c1 (L ). To see
this, it suffices to consider the universal example of a space with two complex line bundles. This is the space
CP∞ × CP∞ . Using the Atiyah-Hirzebruch spectral sequence, we get an isomorphism E ∗ (CP∞ × CP∞ ) 'E ∗ (∗)[[u, v]]. Here u and v denote the pullbacks of t ∈ E 2 (CP∞ ) along the two projection maps π1 , π2 :CP∞ × CP∞ → CP∞ ; in other words, we can identify u and v with the Chern classes of the universal linebundles π1∗ O(1) and π2∗ O(1) on CP∞ × CP∞ . We also have a third line bundle O = π1∗ O(1) ⊗ π2∗ O(1). Then1∞∞∗∗
cE1 (O) = f (u, v) ∈ E (CP × CP ) ' E [[u, v]]. By the method of the universal example, we deduce that0EEE 0
c1 (L ⊗ L ) ' f (c1 L, c1 L ) for any pair of line bundles L and L0 on any space X (here we must take somecare with the meaning of this statement, since f is a power series and not a polynomial in general).

Example 2. For the usual theory of Chern classes, f is given by the formula f (u, v) = u + v.
What can we say about the power series f ? In general it is a power series rather than a polynomial, andcan be quite complicated. However, it is not arbitrary: it satisfies certain identities, which reflect the ideathat the tensor product of complex line bundles is commutative and associative up to isomorphism. Moreprecisely, we havef (u, 0) = u = f (0, u)
f (u, v) = f (v, u)f (u, f (v, w)) = f (f (u, v), w).
In general, if R is a commutative ring, then a power series f (u, v) ∈ R[[u, v]] satisfying the identities above
is called a formal group law over R.
We can summarize our discussion as follows: every complex-oriented cohomology theory E determines
a formal group law over the commutative ring E even (∗). This assignment fits into the general paradigm ofalgebraic topology. A cohomology theory E should be regarded as a topological object: it can be represented
by a spectrum, which is a variation on the notion of a space. To this cohomology theory we assign an algebraic
object: a formal group law over a commutative ring. This assignment satisfies both of the requirements
posited at the beginning of this lecture:(a) Though somewhat more complicated than an abelian group or a vector space, a formal group law is
a reasonably tractable mathematical object. In particular, formal group laws have been thoroughly
studied by algebraic geometers and number theorists.
(b) The formal group law associated to a complex-oriented cohomology theory E remembers a great deal
about E. In fact, one can often reconstruct E from its formal group law.
To elaborate on these points, we first note that there is a universal example of a formal group law. Thatis, there is a commutative ring L and a formal group law f (u, v) ∈ L[[u, v]] which is “maximally complicated”,
in the sense that any other formal group law over a commutative ring R is obtained from f (u, v) by meansof a ring homomorphism L → R. The ring L is called the Lazard ring in honor of Lazard, who proved that
L is a polynomial ring (in infinitely many generators).
According to a theorem of Quillen, the Lazard ring L has another incarnation: it is the coefficient ringof the cohomology theory M U of complex bordism (which is universal among complex-oriented cohomology
theories). One can attempt to use this observation to construct an “inverse” to the above constructions.
Namely, suppose we are given a commutative ring R and a formal group law f (u, v) ∈ R[[u, v]], classified bya map L → R. We can then attempt to define a new cohomology theory E (having coefficient ring R) bythe formula E ∗ (X) ' M U ∗ (X) ⊗L R for finite complexes X (for this to be sensible, R should be equippedwith a suitable grading; we will suppress mention in the discussion which follows). This construction does
not always work: that is, E ∗ does not always have the excision and Mayer-Vietoris exact sequences thatare required of cohomology theories. However, a fundamental result of Landweber gives a purely algebraic
criterion on φ which, if satisfied, guarantees that E ∗ is a cohomology theory. One can use this criterion toproduce many interesting examples of cohomology theories.
Example 3. One can take R to be the ring of integers and f (u, v) to be the multiplicative formal group
given by f (u, v) = u + v + uv. In this case, Landweber’s theorem applies and produces a cohomology theory,
namely, complex K-theory.

2Motivated by Example 3, it is natural to ask what other cohomology theories can be produced by meansof Landweber’s theorem: that is, starting with a map of affine schemes φ : Spec R → Spec L = A∞ . First,
we should note that the map φ is not really fundamental. The formal group law associated to a cohomology
theory E depends not only on E, but also on a choice of complex orientation t ∈ E 2 (CP∞ ). The collectionof all such choices is acted on by the group G of coordinate changes
t 7→ t + a1 t2 + a2 t3 + ...
Consequently, our real interest is not in the moduli space Spec L of formal group laws, but in the quotientSpec L/G. This is a kind of algebraic stack, called the moduli stack of formal groups (more precisely, it is themoduli stack of formal groups with trivialized Lie algebras). The main thrust of this course can be statedas follows:• The structure of the stable homotopy category is controlled by the geometry of the stack Spec L/G.
For example, every complex-orientable cohomology theory E determines a commutative ring R = E even (∗)
and a formal group over R, which we can think of as a map Spec R → Spec L/G. This construction provides
the beginning of a rough dictionary:Multiplicative cohomology theories
Cohomology theories
Complex bordism

Affine schemes over Spec L/G
Quasi-coherent sheaves on Spec L/GModuli Space Spec L ' A∞ of formal group laws

···

···

As we will see over the course of the semester, these ideas give an extremely useful picture of the stablehomotopy category.

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