nLab complex oriented cohomology theory





Special and general types

Special notions


Extra structure



Complex geometry



A complex oriented cohomology theory is a Whitehead-generalized 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.


In terms of generalized first Chern classes

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

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

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

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

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

By the suspension isomorphism there is an identification

E˜ 2(S 2)E˜ 0(S 0)E 0(*)π 0(E) \tilde E^2(S^2) \simeq \tilde E^0(S^0) \simeq E^0(\ast) \simeq \pi_0(E)

with the commutative ring underlying EE. Write 1π 0(E)1 \in \pi_0(E) for the multiplicative identity element in this ring.


(complex oriented cohomology theory)

A multiplicative cohomology theory EE is complex orientable if the following equivalent conditions hold:

  1. The morphism

    i *:E 2(BU(1))E 2(S 2) i^\ast \;\colon\; E^2(B U(1)) \longrightarrow E^2(S^2)

    is surjective.

  2. The morphism on reduced cohomology

    i˜ *:E˜ 2(BU(1))E˜ 2(S 2)π 0(E) \tilde i^\ast \;\colon\; \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2) \simeq \pi_0(E)

    is surjective.

  3. The ring unit 1π 0(E)1 \in \pi_0(E) is in the image of the morphism i˜ *\tilde i^\ast.

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

c 1 EE˜ 2(BU(1)) c_1^E \in \tilde E^2(B U(1))

(the “first generalized Chern class”) such that

i *c 1 E=1π 0(E). i^\ast c^E_1 = 1 \in \pi_0(E) \,.

Since BU(1)K(,2)B U(1) \simeq K(\mathbb{Z},2) is the classifying space for complex line bundles, it follows that a complex orientation on E E^\bullet induces an EE-generalization of the first Chern class which to a complex line bundle \mathcal{L} on XX, classified by ϕ:XBU(1)\phi \colon X \to B U(1) assigns the class c 1()ϕ *c 1 Ec_1(\mathcal{L}) \coloneqq \phi^\ast c_1^E. This construction extends to a general construction of EE-Chern classes (Conner-Floyd Chern classes).


(complex EE-orientation by extensions and their obstructions)

In terms of classifying maps, Def. means that a complex orientation c 1 Ec_1^E in EE-cohomology theory is equivalently an extension (in the classical homotopy category) of the map Σ 21:P 1Ω 2E\Sigma^2 1 \,\colon\, \mathbb{C}P^1 \longrightarrow \Omega^{\infty-2} E (which classifies the suspended identity in the cohomology ring) along the canonical inclusion of complex projective spaces

(1)P 1 Σ 21 E Ω 2E c 1 E P \array{ \mathbb{C}P^1 & \overset{ \Sigma^2 1_E }{ \longrightarrow } & \Omega^{\infty - 2} E \\ \big\downarrow & \nearrow \mathrlap{ {}_{c_1^E} } \\ \mathbb{C}P^\infty }

Notice that the complex projective spaces form a cotower

*=P 0P 1P 2P 3P =limP \ast \,=\, \mathbb{C}P^0 \hookrightarrow \mathbb{C}P^1 \hookrightarrow \mathbb{C}P^2 \hookrightarrow \mathbb{C}P^3 \hookrightarrow \cdots \hookrightarrow \mathbb{C}P^\infty \,=\, \underset{\longrightarrow}{\lim} \mathbb{C}P^\bullet

where each inclusion stage is (by this Prop., see at cell structure of projective spaces) the coprojection of a pushout of topological spaces (or rather: of pointed topological spaces) of the form

D 2n+2 P n+1 (po) S 2n+1 h 2n+1 P n \array{ D^{2n+2} & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(po)}}}& \big\uparrow \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n }

(where h 2n+1h^{2n+1}_{\mathbb{C}} is the complex Hopf fibration in dimension 2n+12n+1) hence of a homotopy pushout of underlying homotopy types (rather: of pointed homotopy types) of this form:

* P n+1 (hpo) S 2n+1 h 2n+1 P n \array{ \ast & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n }

Therefore, a complex orientation by extension (1) is equivalently the homotopy colimiting map of a sequence

(Σ 21=c 1 E,0,c 1 E,1,c 1 E,2,) \big( \Sigma^2 1 \,=\, c_1^{E,0} ,\, c_1^{E,1} ,\, c_1^{E,2} ,\, \cdots \big)

of finite-stage extensions

* P n+1 c 1 E,n+1 Ω 2E (hpo) c 1 E,n S 2n+1 h 2n+1 P n. \array{ \ast & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} & \overset{ c_1^{E,n+1} }{\longrightarrow} & \Omega^{\infty -2} E \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow & \nearrow \mathrlap{ {}_{c_1^{E,n}} } \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n \,. }

Moreover, by the defining universal property of the homotopy pushout, the extension c 1 E,n+1c_1^{E,n+1} of c 1 E,nc_1^{E,n} is equivalently a choice of homotopy which trivializes the pullback of c 1 E,nc_1^{E,n} to the 2n+1-sphere:

* Ω 2E c 1 E,n+1 c 1 E,n S 2n+1 h 2n+1 P n. \array{ \ast & \overset{}{\longrightarrow} & \Omega^{\infty - 2} E \\ \big\uparrow & {}_{ c_1^{E,n+1} } \seArrow & \big\uparrow \mathrlap{ ^{_{ c_1^{E,n} }} } \\ S^{2n+1} &\underset{ h^{2n+1}_{\mathbb{C}} }{\longrightarrow}& \mathbb{C}P^n \,. }

This means, first of all, that the non-triviality of the pullback class

(h 2n+1) *(c 1 E,n)E˜ 2(S 2n+1)E 2n1 \big( h^{2n+1}_{\mathbb{C}} \big)^\ast ( c_1^{E,n} ) \;\in\; \widetilde E^2 \big( S^{2n+1} \big) \;\simeq\; E_{2n-1}

is the obstruction to the existence of the extension/orientation at this stage.

It follows that if these obstructions all vanish, then a complex EE-orientation does exist.

A sufficient condition for this is, evidently, that the reduced EE-cohomology of all odd-dimensional spheres vanishes, hence, that the graded EE-cohomology ring E E_\bullet is trivial in odd degrees (an even cohomology theory).

(see also Lurie, Lecture 6, Remark 4)

In terms of orientations of fibers of complex vector bundles


(complex EE-orientations from first EE-Chern class)

To see that a generalized Chern class c 1 Ec^E_1 in the above sense indeed has to do with complex orientations, namely of, first of all, fibers of complex line bundles, notice that any complex projective space is equivalently the Thom space of the dual tautological line bundle (see there) on the complex projective space of one complex dimension lower:

The morphism of E-cohomology Milnor sequences that this transformation induces shows (see the proof of Prop. for details) that pullback along the zero section identifies the EE-cohomology of the Thom space of the universal complex line bundle with that of the base classifying space:

(In fact the zero-section of the Thom space of the universal complex line bundle is itself a weak homotopy equivalence, see this Lemma.)

But this means that the EE-Chern class c 1 Ec^E_1 has a unique pre-image which is an EE-Thom class th 1 Eth^E_1 on the universal complex line bundle, and hence pulls back to an EE-Thom class on any complex line bundle.

In fact more is true: The choice of c 1 Ec^E_1 induces a sequence of universal Conner-Floyd Chern classes c n Ec^E_n for all nn \in \mathbb{N}, and by an elaboration of the previous argument these are pullbacks along the zero-section of universal EE-Thom classes on the universal complex vector bundle of any rank nn (see this Lemma).

In terms of genera

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


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

(Hopkins 99, section 4, Lurie, lecture 6, theorem 8)

See at universal complex orientation on MU.


One might hope that the above universal property can be refined to say that that if EE is an E E_\infty-ring spectrum, then complex orientations of EE are in bijection with E E_\infty ring maps MUEMU \to E. This is not the case; Ando classified H H_\infty ring maps MUEMU \to E in his thesis and in particular showed that not every complex orientation gives rise to an H H_\infty ring map out of MUMU. More recently, Hopkins and Lawson have classified the further structure constituting an E E_\infty ring map out of MUMU.


Examples of complex orientable cohomology theories:


(ordinary cohomology)

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

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

defines a complex orientation of HH\mathbb{Z}.


(topological K-theory)

For E=KUE = KU complex topological K-theory, then the class of the image of the universal complex line bundle 𝒪(1)\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.


(complex cobordism)

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

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

defines a complex orientation.


Brown-Peterson cohomologyE=BP E = B P^\bullet.


Cohomology ring of BU(1)B U(1)


Given a complex oriented cohomology theory (E ,c 1 E)(E^\bullet, c^E_1) according to def. , then there are isomorphisms of graded rings

  1. E (BU(1))E (*)[[c 1 E]]E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]

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

  2. E (BU(1)×BU(1))E (*)[[c 1 E1,1c 1 E]]E^\bullet(B U(1) \times B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E \otimes 1 , 1 \otimes c_1^E ] ].


We may realize the classifying space BU(1)B U(1) as the infinite complex projective space P =lim nP n\mathbb{C}P^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n (exmpl.). There is a standard CW-complex-structure on the classifying space P \mathbb{C}P^\infty, given by inductively identifying P n+1\mathbb{C}P^{n+1} with the result of attaching a single 2n2n-cell to P n\mathbb{C}P^n (prop.). With this structure, the unique 2-cell inclusion i:S 2P i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty is identified with the canonical map S 2BU(1)S^2 \to B U(1).

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

H (P n,E (*))E (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 (P n,)[c 1]/((c 1) n+1), H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n+1}) \,,

where c 1c_1 represents a unit in H 2(S 2,)H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}, and since similarly the ordinary homology of P n\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(P n),E (*))=0Ext^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 (P n,E (*))E (*)[c 1]/(c 1 n+1). 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 1c_1 is identified with a generator of the relative cohomology

E 2((P n) 2,(P n) 1)E˜ 2(S 2) 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 2S^2 is the unique 2-cell of P n\mathbb{C}P^n in the standard cell model).

This means that c 1c_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(P n)E^2(\mathbb{C}P^n) and hence precisely if there exists a complex orientation c 1 Ec_1^E on EE. Since this is the case by assumption on EE, c 1c_1 is a permanent cocycle. (For the fully detailed argument, see (Pedrotti 16)).

The same argument applied to all elements in E (*)[c]E^\bullet(\ast)[c], or else the E (*)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 (*)[c 1]/(c 1 n+1)E^\bullet(\ast) [c_1] / (c_1^{n+1}), hence that the given AHSS collapses on the second page to give

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

or in more detail:

p,{E (*) ifp2nandeven 0 otherwise. \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 p,\mathcal{E}_\infty^{p,\bullet} are free modules over E (*)E^\bullet(\ast), and since the filter stage inclusions F p+1E (X)F pE (X)F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X) are E (*)E^\bullet(\ast)-module homomorphisms (prop.) the extension problem trivializes, in that all the short exact sequences

0F p+1E p+(X)F pE p+(X) p,0 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 E (*) 1( p,,)=0Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0 vanishes on the free module, hence projective module p,\mathcal{E}_\infty^{p,\bullet}).

In conclusion, this gives an isomorphism of graded rings

E (P n)p p,E (*)[c 1]/((c 1 E) n+1). 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 ((P ) 2n+2)=E (P n+1)E (P n)=E ((P ) 2n) 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:

E (BU(1)) E (P ) E (lim nP n) lim nE (P n) lim n(E (*)[c 1 E]/((c 1 E) n+1)) E (*)[[c 1 E]], \begin{aligned} E^\bullet(B U(1)) & \simeq E^\bullet(\mathbb{C}P^\infty) \\ & \simeq E^\bullet( \underset{\longrightarrow}{\lim}_n \mathbb{C}P^n ) \\ &\simeq \underset{\longleftarrow}{\lim}_n E^\bullet(\mathbb{C}P^n) \\ &\simeq \underset{\longleftarrow}{\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..


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

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

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

of abelian groups, together with a system of group homomorphisms

E n 1(X)E n 2(X)E n 1+n 2(X), E^{n_1}(X) \otimes E^{n_2}(X) \longrightarrow E^{n_1 + n_2}(X) \,,

one for each pair (n 1,n 2)×(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)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

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

while in the latter case it is the Cartesian product

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

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

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

and the product operation is given by

E 2n 1(X)E 2n 2(X)E 2(n 1+n 2)(X) E^{2{n_1}}(X)\otimes E^{2 n_2}(X) \longrightarrow E^{2(n_1 + n_2)}(X)

for all n 1,n 2n_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)

nE n(X)[c 1]. \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

nE n(X)[[c 1]]. \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 P \mathbb{C}P^\infty is naturally computed as the inverse limit of the cohomolgies of the P n\mathbb{C}P^n, each of which unambiguously has the ring structure [c 1]/((c 1) n+1)\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.


(topological K-theory of classifying space of circle group) For the case of topological K-theory E=E = KU, the cohomology ring KU (BU(1))[[c 1 KU]]KU^\bullet(B U(1)) \;\simeq\; \mathbb{Z}[ [ c_1^{KU}] ] (Prop. ) may also be computed via the Atiyah-Segal completion theorem, see the Example there.

Formal group law

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


There is a continuous function

μ:P ×PP \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 XX any paracompact Hausdorff space (notably a CW-complex, since all CW-complexes are paracompact Hausdorff spaces), then

[X,P ×P ] LineBund(X) /×LineBund(X) / [X,μ] [X,P ] LineBund(X) /, \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 LineBund(X) /\mathbb{C}LineBund(X)_{/\sim} denotes the set of isomorphism classes of complex line bundles on XX.

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


By the Yoneda lemma (the fully faithfulness of the Yoneda embedding) there exists such a morphism P ×P P \mathbb{C}P^\infty \times \mathbb{C}P^\infty \longrightarrow \mathbb{C}P^\infty in the classical homotopy category. But since P \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 LineBund(X) /\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 *P \ast \to \mathbb{C}P^\infty) being the neutral element, and that this group structure is natural in XX.


The space BU(1)P 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.


Let (E,c 1 E)(E, c_1^E) be a complex oriented cohomology theory. Under the identification

E (P )π (E)[[c 1 E]],E (P ×P )π (E)[[c 1 E1,1c 1 E]] 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

π (E)[[c 1 E]]E (P )E (P ×P )π (E)[[c 1 E1,1c 1 E]] \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 EE-cohomology along the maps from lemma constitutes a 1-dimensional graded-commutative formal group law (exmpl.) over the graded commutative ring π (E)\pi_\bullet(E) (prop.). If we consider c 1 Ec_1^E to be in degree 2, then this formal group law is compatibly graded.


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


That the grading of c 1 Ec_1^E in prop. is in negative degree is because by definition

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


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


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

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

MU MU^\bullet is the Lazard ring.

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


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

Cohomology ring of BU(n)B U(n)


For EE a complex oriented cohomology theory and nn \in \mathbb{N}, restriction along the canonical map

(BU(1)) nBU(n) (B U(1))^n \longrightarrow B U(n)

induces an isomorphism

E (BU(n))(π E)[[c 1 E,,c n E]]E ((BU(1)) n) Σ nE ((BU(1)) n)(π E)[[(c 1 E) 1,(c 1 E) n]], 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 (BU(n))E^\bullet(B U(n)) with the cyclic group-invariants in E ((BU(1)) n)E^\bullet((B U(1))^n), hence with the power series ring in the elementary symmetric polynomials c i Ec_i^E (the generalized Chern classes) in the c 1 Ec_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.


For EE any cohomology theory and nn \in \mathbb{N}, n1n \geq 1, there is a canonical isomorphism of relative cohomology

E (BU(n),BU(n1))E (B(ζ n),S(ζ n)), E^\bullet(B U(n), B U(n-1)) \simeq E^\bullet( B( \zeta_n), S( \zeta_n) ) \,,

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


Observe that the sphere bundle S(ζ n)BU(n)S(\zeta_n) \to B U(n) of the universal complex vector bundle is equivalently the canonical map BU(n1)BU(n)B U(n-1) \to B U(n).

This follows form the fact that S 2n1U(n)/U(n1)S^{2n-1} \simeq U(n)/U(n-1) and that hence the unit sphere bundle is equivalently the quotient of the U(n)U(n)-universal principal bundle by U(n1)U(n-1)

U(n) * BU(n)S 2n1 U(n)/U(n1) BU(n1) BU(n). \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(ζ n)B(\zeta_n) is weakly equivalent to BU(n)B U(n), and under this identification the map S(ζ n)B(ζ n)S(\zeta_n) \to B(\zeta_n) is equivalent to BU(n1)BU(n)B U(n-1) \to B U(n).


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

(e.g. Lurie 10, lecture 5, prop. 6)

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory



Textbook accounts:

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-.

A comparison between complex orientations and H H_\infty ring maps out of MUMU was given in

  • Matthew Ando, Operations in complex-oriented cohomology theories related to subgroups of formal groups (PhD Thesis)

More recent developments include

Refinement to equivariant complex oriented cohomology theory:

Finite-dimensional complex orientation and Ravenel’s spectra

Discussion of complex orientation (in Whitehead generalized cohomology) on (only) those complex vector bundles which are pulled back from base spaces of bounded cell-dimension (Hopkins 84, 1.2, Ravenel 86, 6.5.2) – or rather, for the most part, of Ravenel's Thom spectra X(n)X(n) and T(m)T(m) (Ravenel 84, Sec. 3) which co-represent these:

Last revised on February 13, 2024 at 06:10:48. See the history of this page for a list of all contributions to it.