Contents

cohomology

complex geometry

# Contents

## Idea

A complex oriented cohomology theory is a generalized (Eilenberg-Steenrod) cohomology theory which is oriented on all complex vector bundles. Examples include ordinary cohomology, complex topological K-theory, elliptic cohomology and cobordism cohomology.

The collection of all complex oriented cohomology theories turns out to be parameterized over the moduli stack of formal group laws. The stratification of this stack by the height of formal group leads to the stratification of complex oriented cohomology theory by “chromatic level”, a perspective also known as chromatic homotopy theory.

For more detailed introduction see at Introduction to Cobordism and Complex Oriented Cohomology.

## Definition

### In terms of generalized first Chern classes

Write $\mathbb{C}P^\infty \simeq B U(1) \simeq K(\mathbb{Z},2)$ for the inifinite complex projective space, equivalently the classifying space for circle group-principal bundles (an Eilenberg-MacLane space); write $S^2$ for the 2-sphere and write

$i \;\colon\; S^2 \longrightarrow B U(1)$

for a representative of $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$, classifying the universal complex line bundle. Regard both $S^2$ and $B U(1)$ as pointed homotopy types and take $i$ to be a pointed morphism.

Let $E^\bullet$ be a multiplicative cohomology theory, i.e. a functor $X \mapsto \pi_\bullet[X,E]$ for $E$ a ring spectrum. Write $\tilde E^\bullet$ for the corresponding reduced cohomology on pointed topological spaces, such that for any pointed space $X$ there is a canonical direct sum decomposition (this prop.)

$E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast) \,.$

By the suspension isomorphism there is an identification

$\tilde E^2(S^2) \simeq \tilde E^0(S^0) \simeq E^0(\ast) \simeq \pi_0(E)$

with the commutative ring underlying $E$. Write $1 \in \pi_0(E)$ for the multiplicative identity element in this ring.

###### Definition

A multiplicative cohomology theory $E$ is complex orientable if the following equivalent conditions hold

1. The morphism

$i^\ast \;\colon\; E^2(B U(1)) \longrightarrow E^2(S^2)$

is surjective.

2. The morphism

$\tilde i^\ast \;\colon\; \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2) \simeq \pi_0(E)$

is surjective.

3. The element $1 \in \pi_0(E)$ is in the image of the morphism $\tilde i^\ast$.

A complex orientation on a multiplicative cohomology theory $E^\bullet$ is an element

$c_1^E \in \tilde E^2(B U(1))$

(the “first generalized Chern class”) such that

$i^\ast c^E_1 = 1 \in \pi_0(E) \,.$
###### Remark

Since $B U(1) \simeq K(\mathbb{Z},2)$ is the classifying space for complex line bundles, it follows that a complex orientation on $E^\bullet$ induces an $E$-generalization of the first Chern class which to a complex line bundle $\mathcal{L}$ on $X$ classified by $\phi \colon X \to B U(1)$ assigns the class $c_1(\mathcal{L}) \coloneqq \phi^\ast c_1^E$. This construction extends to a general construction of $E$-Chern classes.

### In terms of genera and $E_\infty$ orientations

Complex orientation in the above sense is indeed universal MU-orientation in generalized cohomology:

###### Proposition

For $E$ a homotopy commutative ring spectrum then there is a bijection between complex orientations of $E$-cohomology and homotopy commutative ring spectrum-homomorphisms $MU \longrightarrow E$ out of MU.

###### Remark

In terms of E-∞ geometry/spectral geometry, prop. says that complex orientation on $E$ is equivalently a morphism

$Spec(E)\longrightarrow Spec(MU)$

exhibiting the affine E-∞ scheme $Spec(E)$ as sitting over $Spec(MU)$. By Quillen's theorem on MU,

## Examples

Examples of complex orientable cohomology theories:

###### Example

(ordinary cohomology)

For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum, the ordinary first Chern class

$c_1 \in H^2(B U(1), \mathbb{Z})$

defines a complex orientation of $H\mathbb{Z}$.

###### Example

(topological K-theory)

For $E = KU$ complex topological K-theory, then the class of the image of the the universal complex line bundle $\mathcal{O}(1)$ in reduced K-theory is a complex orientation.

The induced formal group law (by prop. ) is the multiplicative formal group law.

For details see at topological K-theory the section Complex orientation and Formal group law.

###### Example

(complex cobordism)

For $E = MU$ complex cobordism cohomology theory, the canonical map

$B U(1) \stackrel{\simeq}{\to} MU(1) \to MU$

defines a complex orientation.

###### Example

Brown-Peterson cohomology$E = B P^\bullet$.

## Properties

### Cohomology ring of $B U(1)$

###### Proposition

Given a complex oriented cohomology theory $(E^\bullet, c^E_1)$ according to def. , then there are isomorphisms of graded rings

1. $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$

(between the $E$-cohomology ring of $B U(1)$ and the formal power series (but see remark ) in one generator of even degree over the $E$-cohomology ring of the point);

2. $E^\bullet(B U(1) \times B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E \otimes 1 , 1 \otimes c_1^E ] ]$.

###### Proof

We may realize the classifying space $B U(1)$ as the infinite complex projective space $\mathbb{C}P^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n$ (exmpl.). There is a standard CW-complex-structure on the classifying space $\mathbb{C}P^\infty$, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$ (prop.). With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.

Then consider the Atiyah-Hirzebruch spectral sequence for the $E$-cohomology of $\mathbb{C}P^n$.

$H^\bullet(\mathbb{C}P^n, E^\bullet(\ast)) \;\Rightarrow\; E^\bullet(\mathbb{C}P^n) \,.$

Since (prop.) the ordinary cohomology with integer coefficients of projective space is

$H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n-1}) \,,$

where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the ordinary homology of $\mathbb{C}P^n$ is a free abelian group (prop.), hence a projective object in abelian groups (prop.), the Ext-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the universal coefficient theorem (prop.) gives that the second page of the spectral sequence is

$H^\bullet(\mathbb{C}P^n, E^\bullet(\ast)) \simeq E^\bullet(\ast)[ c_1 ] / (c_1^{n+1}) \,.$

By the standard construction of the Atiyah-Hirzebruch spectral sequence (here) in this identification the element $c_1$ is identified with a generator of the relative cohomology

$E^2((\mathbb{C}P^n)_2, (\mathbb{C}P^n)_1) \simeq \tilde E^2(S^2)$

(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).

This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument, see (Pedrotti 16)).

The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials (prop.), implies that all these elements are permanent cocycles.

Since the AHSS of a multiplicative cohomology theory is a multiplicative spectral sequence (prop.) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give

$\mathcal{E}_\infty^{\bullet,\bullet} \simeq E^\bullet(\ast)[ c_1^{E} ] / ((c_1^E)^{n+1})$

or in more detail:

$\mathcal{E}_\infty^{p,\bullet} \simeq \left\{ \array{ E^\bullet(\ast) & \text{if}\; p \leq 2n \; and\; even \\ 0 & otherwise } \right. \,.$

Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are free modules over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-module homomorphisms (prop.) the extension problem trivializes, in that all the short exact sequences

$0 \to F^{p+1}E^{p+\bullet}(X) \longrightarrow F^{p}E^{p+\bullet}(X) \longrightarrow \mathcal{E}_\infty^{p,\bullet} \to 0$

split (since the Ext-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the free module, hence projective module $\mathcal{E}_\infty^{p,\bullet}$).

In conclusion, this gives an isomorphism of graded rings

$E^\bullet(\mathbb{C}P^n) \simeq \underset{p}{\oplus} \mathcal{E}_\infty^{p,\bullet} \simeq E^\bullet(\ast)[ c_1 ] / ((c_1^{E})^{n+1}) \,.$

A first consequence is that the projection maps

$E^\bullet((\mathbb{C}P^\infty)_{2n+2}) = E^\bullet(\mathbb{C}P^{n+1}) \to E^\bullet(\mathbb{C}P^{n+}) = E^\bullet((\mathbb{C}P^\infty)_{2n})$

are all epimorphisms. Therefore this sequence satisfies the Mittag-Leffler condition (def., exmpl.) and therefore the Milnor exact sequence for generalized cohomology (prop.) finally implies the claim:

\begin{aligned} E^\bullet(B U(1)) & \simeq E^\bullet(\mathbb{C}P^\infty) \\ & \simeq E^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n ) \\ &\simeq \underset{\longrightarrow}{\lim}_n E^\bullet(\mathbb{C}P^n) \\ &\simeq \underset{\longrightarrow}{\lim}_n ( E^\bullet(\ast) [c_1^E] / ((c_1^E)^{n+1}) ) \\ & \simeq E^\bullet(\ast)[ [ c_1^E ] ] \,, \end{aligned}

where the last step is this prop..

###### Remark

There is in general a choice to be made in interpreting the cohomology groups of a multiplicative cohomology theory $E$ as a ring:

a priori $E^\bullet(X)$ is a sequence

$\{E^n(X)\}_{n \in \mathbb{Z}}$

of abelian groups, together with a system of group homomorphisms

$E^{n_1}(X) \otimes E^{n_2}(X) \longrightarrow E^{n_1 + n_2}(X) \,,$

one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.

In turning this into a single ring by forming formal sums of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.

In the former case the ring obtained is the direct sum

$\oplus_{n \in \mathbb{N}} E^n(X)$

while in the latter case it is the Cartesian product

$\prod_{n \in \mathbb{N}} E^n (X) \,.$

These differ in general. For instance if $E$ is ordinary cohomology with integer coefficients and $X$ is infinite complex projective space $\mathbb{C}P^\infty$, then (prop.)

$E^n(X) = \left\{ \array{ \mathbb{Z} & n \; even \\ 0 & otherwise } \right.$

and the product operation is given by

$E^{2{n_1}}(X)\otimes E^{2 n_2}(X) \longrightarrow E^{2(n_1 + n_2)}(X)$

for all $n_1, n_2$ (and zero in odd degrees, necessarily). Now taking the direct sum of these, this is the polynomial ring on one generator (in degree 2)

$\oplus_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z}[c_1] \,.$

But taking the Cartesian product, then this is the formal power series ring

$\prod_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z} [ [ c_1 ] ] \,.$

A priori both of these are sensible choices. The former is the usual choice in traditional algebraic topology. However, from the point of view of regarding ordinary cohomology theory as a multiplicative cohomology theory right away, then the second perspective tends to be more natural;

The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the inverse limit of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the category of commutative rings right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the formal power series ring (see here).

See also for instance remark 1.1. in Jacob Lurie: A Survey of Elliptic Cohomology.

### Formal group law

Let again $B U(1)$ be the classifying space for complex line bundles, modeled, in particular, by infinite complex projective space $\mathbb{C}P^\infty)$.

###### Lemma

There is a continuous function

$\mu \;\colon\; \mathbb{C}P^\infty \times \mathbb{C}P \longrightarrow \mathbb{C}P^\infty$

which represents the tensor product of line bundles in that under the defining equivalence, and for $X$ any paracompact Hausdorff space (notably a CW-complex, since all CW-complexes are paracompact Hausdorff spaces), then

$\array{ [X, \mathbb{C}P^\infty \times \mathbb{C}P^\infty] &\simeq& \mathbb{C}LineBund(X)_{/\sim} \times \mathbb{C}LineBund(X)_{/\sim} \\ {}^{\mathllap{[X,\mu]}}\downarrow && \downarrow^{\mathrlap{\otimes}} \\ [X,\mathbb{C}P^\infty] &\simeq& \mathbb{C}LineBund(X)_{/\sim} } \,,$

where $[-,-]$ denotes the hom-sets in the (Serre-Quillen-)classical homotopy category and $\mathbb{C}LineBund(X)_{/\sim}$ denotes the set of isomorphism classes of complex line bundles on $X$.

Together with the canonical point inclusion $\ast \to \mathbb{C}P^\infty$, this makes $\mathbb{C}P^\infty$ an abelian group object in the classical homotopy category (an abelian H-group).

###### Proof

By the Yoneda lemma (the fully faithfulness of the Yoneda embedding) there exists such a morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \longrightarrow \mathbb{C}P^\infty$ in the classical homotopy category. But since $\mathbb{C}P^\infty$ admits the structure of a CW-complex (prop.) it is cofibrant in the standard model structure on topological spaces, as is its Cartesian product with itself (prop.). Since moreover all spaces are fibrant in the classical model structure on topological spaces, it follows (by this lemma) that there is an actual continuous function representing that morphism in the homotopy category.

That this gives the structure of an abelian group object now follows via the Yoneda lemma from the fact that each $\mathbb{C}LineBund(X)_{/\sim}$ has the structure of an abelian group under tensor product of line bundles, with the trivial line bundle (wich is classified by maps factoring through $\ast \to \mathbb{C}P^\infty$) being the neutral element, and that this group structure is natural in $X$.

###### Remark

The space $B U(1) \simeq \mathbb{C}P^\infty$ has in fact more structure than that of an H-group from lemma . As an object of the homotopy theory represented by the classical model structure on topological spaces, it is a 2-group, a 1-truncated infinity-group.

###### Proposition

Let $(E, c_1^E)$ be a complex oriented cohomology theory. Under the identification

$E^\bullet(\mathbb{C}P^\infty) \simeq \pi_\bullet(E)[ [ c^E_1 ] ] \;\;\;\,, \;\;\; E^\bullet(\mathbb{C}P^\infty \times \mathbb{C}P^\infty) \simeq \pi_\bullet(E)[ [ c^E_1 \otimes 1 , \, 1 \otimes c^E_1 ] ]$

from prop. , the operation

$\pi_\bullet(E) [ [ c^E_1 ] ] \simeq E^\bullet(\mathbb{C}P^\infty) \longrightarrow E^\bullet( \mathbb{C}P^\infty \times \mathbb{C}P^\infty ) \simeq \pi_\bullet(E)[ [ c_1^E \otimes 1, 1 \otimes c_1^E ] ]$

of pullback in $E$-cohomology along the maps from lemma constitutes a 1-dimensional graded-commutative formal group law (exmpl.) over the graded commutative ring $\pi_\bullet(E)$ (prop.). If we consider $c_1^E$ to be in degree 2, then this formal group law is compatibly graded.

###### Proof

The associativity and commutativity conditions follow directly from the respective properties of the map $\mu$ in lemma . The grading follows from the nature of the identifications in prop. .

###### Remark

That the grading of $c_1^E$ in prop. is in negative degree is because by definition

$\pi_\bullet(E) = E_\bullet = E^{-\bullet}$

(rmk.).

Under different choices of orientation, one obtains different but isomorphic formal group laws.

###### Example

The formal group law of complex cobordism cohomology theory, example is universal in that for every commutative ring $R$ there is a natural bijection

$CRing(MU^\bullet, R) \simeq FormalGroupLaws_{/R} \,.$

$MU^\bullet$ is the Lazard ring.

This is Milnor-Quillen's theorem on MU (involving Lazard's theorem).

###### Example

The formal group law of Brown-Peterson cohomology theory, example is universal for $p$-local cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, p-typical formal group laws.

### Cohomology ring of $B U(n)$

###### Proposition

For $E$ a complex oriented cohomology theory and $n \in \mathbb{N}$, restriction along the canonical map

$(B U(1))^n \longrightarrow B U(n)$

induces an isomorphism

$E^\bullet(B U(n)) \stackrel{\simeq}{\longrightarrow} (\pi_\bullet E)[ [ c^E_1, \cdots, c^E_n ] ] \simeq E^\bullet((B U(1))^n)^{\Sigma_n} \hookrightarrow E^\bullet((B U(1))^n) \simeq (\pi_\bullet E)[ [(c_1^E)_1, \cdots (c_1^E)_n ] ] \,,$

of $E^\bullet(B U(n))$ with the cyclic group-invariants in $E^\bullet((B U(1))^n)$, hence with the power series ring in the elementary symmetric polynomials $c_i^E$ (the generalized Chern classes) in the $c_1^E$-s (the generalized first Chern classes of prop. ).

Use this proposition to reduce to the situation for ordinary Chern classes. (e.g. Lurie 10, lecture 4)

### Canonical orientation on complex vector bundles

The follows says that complex oriented cohomology theories in the sense of def. , indeed canonically have an orientation in generalized cohomology for the (spherical fibration of) any complex vector bundle.

For more details see at universal complex orientation on MU.

###### Proposition

For $E$ any cohomology theory and $n \in \mathbb{N}$, $n \geq 1$, there is a canonical isomorphism of relative cohomology

$E^\bullet(B U(n), B U(n-1)) \simeq E^\bullet( B( \zeta_n), S( \zeta_n) ) \,,$

where $\zeta_n \coloneqq E U(n) \underset{U(n)}{\times} \mathbb{R}^{2n}$ is the universal complex vector bundle.

###### Proof

Observe that the sphere bundle $S(\zeta_n) \to B U(n)$ of the universal complex vector bundle is equivalently the canonical map $B U(n-1) \to B U(n)$.

This follows form the fact that $S^{2n-1} \simeq U(n)/U(n-1)$ and that hence the unit sphere bundle is equivalently the quotient of the $U(n)$-universal principal bundle by $U(n-1)$

$\array{ U(n) &\longrightarrow& \ast \\ && \downarrow \\ && B U(n) } \;\;\;\; \stackrel{}{\mapsto}\;\;\;\; \array{ S^{2n-1} \simeq & U(n)/U(n-1) &\longrightarrow& B U(n-1) \\ & && \downarrow \\ & && B U(n) } \,.$

The unit ball bundle $B(\zeta_n)$ is weakly equivalent to $B U(n)$, and under this identification the map $S(\zeta_n) \to B(\zeta_n)$ is equivalent to $B U(n-1) \to B U(n)$.

###### Proposition

For $E$ a complex oriented cohomology theory, its $n$th generalized Chern class $c^E_n$, prop. , identified as an element of $E^\bullet(B(\zeta_n), S(\zeta_n))$ via prop. , is a Thom class.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

## References

Textbook accounts include

Introduction includes

The perspective of chromatic homotopy theory originates in

and is further developed in

See also the references at equivariant cohomology – References – Complex oriented cohomology-.

More recent developments include

Last revised on June 6, 2017 at 08:12:03. See the history of this page for a list of all contributions to it.