Let ${\Lambda}$ be a ${\mathbb{Q}}$-algebra. A genus is a homomorphism

$\displaystyle \phi: \Omega_{SO} \rightarrow \Lambda,$

where ${\Omega_{SO} }$ is the oriented cobordism ring. In other words, a genus ${\phi}$ assigns to every compact, oriented manifold ${M}$ an element ${\phi(M) \in \Lambda}$. This satisfies the conditions:

1. ${\phi(M \sqcup M') = \phi(M) + \phi(M')}$.
2. ${\phi(M \times M') = \phi(M) \phi(M')}$.
3. ${\phi(\partial N) =0 }$ for any manifold-with-boundary ${N}$.

A fundamental example of a genus is the signature ${\sigma}$, which assigns to every manifold ${M}$ of dimension ${4k}$ the signature of the quadratic form on ${H^{2k}(M; \mathbb{R})}$. (Also, ${\sigma}$ is zero on manifolds whose dimension is not divisible by four.)

Before moving on, I’d like to work out the analog for real-oriented cohomology theories (which is tangential to the rest of the story, though). This is considerably less interesting, but perhaps it’s a toy example of the ideas explained in the last few posts without the full-blown machinery of the Adams spectral sequence and so forth.

So, let’s state what the analogous ideas are in the real-oriented context:

1. real-oriented ring spectrum ${E}$ is a ring spectrum together with a functorial, multiplicative choice of Thom classes for real vector bundles; equivalently, there is a morphism of ring spectra

$\displaystyle MO \rightarrow E.$

That is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

2. If ${E}$ is real-oriented, then ${\pi_* E}$ is a ${\mathbb{Z}/2}$-vector space, and all the usual computations of ${H_*( BO; \mathbb{Z}/2)}$ and so forth work just fine for ${E}$, and there is a theory of Stiefel-Whitney classes in ${E}$-cohomology.
3. Given a real-oriented spectrum ${E}$, we thus have ${E^*(\mathbb{RP}^\infty) = \pi_* E [[t]]}$ where ${t}$ is the Stiefel-Whitney class of the tautological bundle. Similiarly, ${E^*(\mathbb{RP}^\infty \times \mathbb{RP}^\infty) = \pi_* E [[t_1, t_2]]}$.

Since ${\mathbb{RP}^\infty}$ is the classifying space for real line bundles, there is a monoidal product

$\displaystyle \mathbb{RP}^\infty \times \mathbb{RP}^\infty \rightarrow \mathbb{RP}^\infty,$

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over ${\pi_* E}$. This formal group law ${f(\cdot, \cdot) \in \pi_* E [[x, y]]}$ has the property that if ${\mathcal{L}_1, \mathcal{L}_2}$ are two line bundles over a finite-dimensional space ${X}$, then in ${E^*(X)}$,

$\displaystyle w_1( \mathcal{L}_1 \otimes \mathcal{L}_2) = f( w_1(\mathcal{L}_1), w_1(\mathcal{L}_2)).$

So far everything has been analogous to the complex-oriented case, but there is an extra feature here which changes the picture drastically. Namely, when one works with realline bundles ${\mathcal{L}}$, we have that ${\mathcal{L}^{\otimes 2}}$ is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law ${f}$ must satisfy

$\displaystyle f(x, x) = 0.$

This is not satisfied by, say, the multiplicative formal group law. (And in fact, ${KO}$-theory—the natural candidate for this—is not real-oriented, only spin-oriented.)

With this in mind, the analog of Quillen’s theorem for ${MU}$ becomes:

Theorem 1 (Quillen) The formal group law for ${MO}$ is the universal formal group law over a satisfying ${f(x, x) = 0}$. (more…)

Our goal is now to return to topology, and in particular to study the formal group law of the universal complex-oriented theory ${MU}$ (complex cobordism). As we computed using the Adams spectral sequence,

$\displaystyle \pi_* MU \simeq \mathbb{Z}[x_1, x_2, \dots ] , \quad \deg x_i = 2i.$

This is the Lazard ring, by the computations of the previous couple of posts. On the other hand, it is not at all clear that the map

$\displaystyle L \rightarrow \pi_* MU$

classifying the formal group law over ${\pi_* MU}$ (arising from the complex orientation) is actually an isomorphism: in other words, that the formal group law of ${MU}$ is the universal one. The fact that it is in fact an isomorphism is the content of Quillen’s theorem, which will be proved in this post. (more…)

This is the third in the series of posts intended to work through the proof of Lazard’s theorem, that the Lazard ring classifying the universal formal group law is actually a polynomial ring on a countable set of generators. In the first post, we reduced the result to an elementary but tricky “symmetric 2-cocycle lemma.” In the previous post, we proved most of the symmetric 2-cocycle lemma, except in characteristic zero. The case of characteristic zero is not harder than the cases we handled (it’s easier), but in this post we’ll complete the proof of that case by exhibiting a very direct construction of logarithms in characteristic zero. Next, I’ll describe an application in Lazard’s original paper, on “approximate” formal group laws.

After this, I’m going to try to move back to topology, and describe the proof of Quillen’s theorem on the formal group law of complex cobordism. The purely algebraic calculations of the past couple of posts will be necessary, though.

1. Formal group laws in characteristic zero

The last step missing in the proof of Lazard’s theorem was the claim that the map

$\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ]$

classifying the formal group law obtained from the additive one by “change-of-coordinates” by the exponential series ${\exp(x) = \sum b_i x^{i+1}}$ is an isomorphism mod torsion. In other words, we have an isomorphism

$\displaystyle L \otimes \mathbb{Q} \simeq \mathbb{Z}[b_1, b_2, \dots ] \otimes \mathbb{Q}.$

In fact, we didn’t really need this: we could have proved the homological 2-cocycle lemma in all cases, instead of just the finite field case, and it would have been easier. But I’d like to emphasize that the result is really something elementary here. In fact, what it is saying is that to give a formal group law over a ${\mathbb{Q}}$-algebra is equivalent to giving a choice of series ${\sum b_i x^{i+1}}$.

Definition 1 An exponential for a formal group law ${f(x,y) }$ is a power series ${\exp(x) = x + b_1 x^2 + \dots}$ such that

$\displaystyle f(x,y) = \exp( \exp^{-1}(x) + \exp^{-1}(y)).$

The inverse power series ${\exp^{-1}(x)}$ is called the logarithm.

That is, a logarithm is an isomorphism of ${f}$ with the additive formal group law.

So another way of phrasing this result is that:

Proposition 2 A formal group law over a ${\mathbb{Q}}$-algebra has a unique logarithm (i.e., is uniquely isomorphic to the additive one). (more…)

We are in the middle of proving an important result of Lazard:

Theorem 1 The Lazard ring ${L}$ over which the universal formal group law is defined is a polynomial ring in variables ${x_1, x_2, \dots, }$ of degree ${2i}$.

The fact that the Lazard ring is polynomial implies a number of results which are not a priori obvious: for instance, it shows that given a surjection of rings ${ A \twoheadrightarrow B}$, then any formal group law on ${B}$ can be lifted to one over ${A}$.

We began the proof of Lazard’s theorem last time: we produced a map

$\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ], \quad \deg b_i = 2i,$

classifying the formal group law obtained from the additive one ${x+y}$ by the “change of coordinates” ${ \exp(x) = \sum b_i x^{i+1}}$. We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of ${\mathbb{Z}[b_1, b_2, \dots ]}$ could be determined completely. I won’t get into the details of this (it was all in the previous post), because the purpose of this post is to prove a result to which we reduced last time.

Let ${A}$ be an abelian group. A symmetric 2-cocycle is a “polynomial” ${P(x,y) \in A[x, y] = A \otimes_{\mathbb{Z}} \mathbb{Z}[x, y]}$ with the properties:

$\displaystyle P(x, y) = P(y,x)$

and

$\displaystyle P(x, y+z) + P(y, z) = P(x,y) + P(x+y, z).$

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring ${\mathbb{Z} \oplus A}$, as we saw last time: in fact, we can think of them as “deformations” of the additive formal group law.

The main lemma which we stated last time was the following:

Theorem 2 (Symmetric 2-cocycle lemma) A homogeneous symmetric 2-cocycle of degree ${n}$ is a multiple of ${\frac{1}{d} ( ( x+y)^n - x^n - y^n )}$ where ${d =1}$ if ${n}$ is not a power of a prime, and ${d = p}$ if ${n = p^k}$.

For a direct combinatorial proof of this theorem, see Lurie’s notes. I want to describe a longer homological proof, which is apparently due to Mike Hopkins and which appears in the COCTALOS notes. The strategy is to interpret these symmetric 2-cocycles as actual cocycles in a cobar complex computing an ${\mathrm{Ext}}$ group. Then, the strategy is to compute this ${\mathrm{Ext}}$ group independently.

This argument is somewhat longer than the combinatorial one, but it has the benefit (for me) of engaging with some homological algebra (which I need to learn more about), as well as potentially generalizing in other directions.  (more…)

After describing the computation of ${\pi_* MU}$, I’d now like to handle the remaining half of the machinery that goes into Quillen’s theorem: the structure of the universal formal group law.

Let ${R}$ be a (commutative) ring. Recall that a formal group law (commutative and one-dimensional) is a power series ${f(x,y) \in R[[x,y]]}$ such that

1. ${f(x,y) = f(y,x)}$.
2. ${f(x, f(y,z)) = f(f(x,y), z)}$.
3. ${f(x,0) = f(0,x) = x}$.

It is automatic from this by a successive approximation argument that there exists an inverse power series ${i(x) \in R[[x]]}$ such that ${f(x, i(x)) = 0}$.

In particular, ${f}$ has the property that for any ${R}$-algebra ${S}$, the nilpotent elements of ${S}$ become an abelian group with addition given by ${f}$.

A key observation is that, given ${R}$, to specify a formal group law amounts to specifying a countable collection of elements ${c_{i,j}}$ to define the power series ${f(x,y) = \sum c_{i,j} x^i y^j}$. These ${c_{i,j}}$ are required to satisfy various polynomial constraints to ensure that the formal group identities hold. Consequently:

Theorem 1 There exists a universal ring ${L}$ together with a formal group law ${f_{univ}(x,y)}$ on ${L}$, such that any FGL ${f}$ on another ring ${R}$ determines a unique map ${L \rightarrow R}$ carrying ${f_{univ} \mapsto f}$. (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…)