One of the really nice pictures in homotopy theory is the “chromatic” one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups). A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen’s theorem on the formal group of $MU$. I didn’t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I’ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes of Haynes Miller.)

1. Descent

Let’s start with some high-powered generalities that I don’t really understand, and then come back to earth. Consider an ${E_\infty}$-ring ${R}$; the most important examples will be ${R = H \mathbb{Z}/2}$ or ${R = MU}$. There is a map of ${E_\infty}$-rings ${S \rightarrow R}$, where ${S}$ is the sphere spectrum.

Let ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a module over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \otimes X}$, where the smash product of spectra is written ${\otimes}$. In fact, we have an adjunction

$\displaystyle \mathrm{Mod}(S) \rightleftarrows \mathrm{Mod}(R)$

between ${R \otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \otimes X}$ together with the “descent data” on ${X}$. The benefit is that while the homotopy groups ${\pi_* X}$ may be intractable, those of ${R \otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.

Let’s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of

$\displaystyle M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B.$

How does one imitate this construction in homotopy? One then has a cosimplicial ${E_\infty}$-ring given by the cobar construction

$\displaystyle R \rightrightarrows R \otimes R \dots .$

The ${\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram

$\displaystyle R \otimes X \rightrightarrows R \otimes R \otimes X \dots .$ (more…)

(This is the second in a series of posts intended for me to try to understand the connection between stable homotopy theory and formal group laws.)

Last time, we introduced the notion of a complex-oriented cohomology theory and saw that we could imitate the classical theory of Chern classes in one such. In this post, I’d like to describe the universal example of a complex-oriented cohomology theory: complex cobordism. This is going to play a very special role in the next few posts.

To unravel this, let’s try to recall what a complex orientation was. It was a choice of Thom classes of complex vector bundles, functorial in the bundle and multiplicative. For starters, let’s focus now on the functoriality. Let ${E}$ be a cohomology theory represented by a spectrum ${E}$. Then since there is a universal ${n}$-dimensional vector bundle ${\zeta_n \rightarrow BU(n)}$, it follows that a functorial choice of Thom classes for ${n}$-dimensional vector bundles is the same as a Thom class for ${\zeta_n}$. So, all we need to give is an element of ${\widetilde{E}^*(T(\zeta_n))}$. If we (and we henceforth do this) normalize things such that the Thom class of the ${n}$-th degree element is in degree ${n}$, then we have to give an element of ${\widetilde{E}^n(T(\zeta_n))}$.

Definition 1 The spectrum ${MU(n)}$ is ${\Sigma^{-2n}T(\zeta_n)}$.

So another way of saying this is that we should have a map of spectra

$\displaystyle MU(n) \rightarrow E.$

There is a map ${S^{2n} \rightarrow T(\zeta_n)}$ which comes from fixing a basepoint in ${BU(n)}$. So, in other words, to give a functorial complex orientation for ${n}$-dimensional complex vector bundles is to give an element of ${\widetilde{E}^n(T(\zeta_n))}$ which restricts to the generator of ${\widetilde{E}^n(S^{2n})}$. (To check that an element is a Thom class, we only need restrict it to one fiber in each connected component of the base.) (more…)

Let ${E}$ be a multiplicative cohomology theory. We say that ${E}$ is complex-oriented if one is given the data of an element ${t \in \widetilde{E}^2(\mathbb{CP}^\infty)}$ which restricts to the canonical generator of ${\widetilde{E}^2(\mathbb{CP}^1) \simeq \widetilde{E}^0(S^0)}$. It turns out that one has a bit more: a complex orientation gives on a functorial, multiplicative choice of Thom classes for complex vector bundles. In fact, this is a perhaps more natural definition of such a theory.

What does this mean? Given a vector bundle ${\zeta \rightarrow X}$, one can form the Thom space ${T(\zeta) = B(\zeta)/S(\zeta)}$: in other words, the quotient of the unit ball bundle ${B(\zeta)}$ in ${\zeta}$ (with respect to a choice of metric) by the unit sphere bundle ${S(\zeta)}$. When ${X}$ is compact, this is just the one-point compactification of ${\zeta}$.

Definition 1 The vector bundle ${\zeta}$ is orientable for a multiplicative cohomology theory ${E}$ if there exists an element ${\theta \in \widetilde{E}^*( T(\zeta)) = E^*(B(\zeta), S(\zeta))}$ which restricts to a generator on each fiberwise ${E^*(B^n, S^{n-1})}$, where ${\dim \zeta = n}$. Such a ${\theta}$ is called a Thom class.

Observe that for each point ${x \in X}$, there is a restriction map ${\widetilde{E}^*(T(\zeta)) \rightarrow E^*(B_x^n, S_x^{n-1})}$ if the dimension of ${\zeta}$ is ${n}$.

The existence of a Thom class implies a Thom isomorphism, as for ordinary homology.

Theorem 2 (Thom isomorphism) A Thom class ${\theta \in \widetilde{E}^*(T(\zeta))}$ induces an isomorphism

$\displaystyle E^*(X) \simeq \widetilde{E}^*(T(\zeta))$

given by cup-product with ${\theta}$.

In the case of ordinary homology, a Thom class is unique (up to sign) if it exists; in general, though, a Thom class is highly non-unique, and an orientation is additional data than simple orientability.

Here are a few basic cases:

1. Any vector bundle is orientable for ${\mathbb{Z}/2}$-cohomology.
2. An oriented (in the usual sense: i.e., the top wedge power is trivial) vector bundle is one oriented for ${\mathbb{Z}}$-cohomology.
3. Complex vector bundles are oriented for ${K}$-theory. We will see this below.
4. Spin bundles are oriented for ${KO}$-theory. An explicit construction of Thom classes can be made, as virtual bundles arising from Clifford modules: this is in Atiyah-Bott-Shapiro’s paper.
5. A trivial bundle is orientable for any cohomology theory (this is rather uninteresting: the Thom space is just a suspension). (more…)