complex oriented cohomology theory in nLab

Context

Cohomology

Complex geometry

• Idea
• Definition
• In terms of generalized first Chern classes
• In terms of orientations of fibers of complex vector bundles
• In terms of genera
• Examples
• Properties
• Cohomology ring of BU(1)B U(1)
• Formal group law
• Cohomology ring of BU(n)B U(n)
• Canonical orientation on complex vector bundles
• Related concepts
• References
• General
• Finite-dimensional complex orientation and Ravenel’s spectra

Definition

(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 1∈π 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 E∈E˜ 2(BU(1)) c_1^E \in \tilde E^2(B U(1))

(the “first generalized Chern class”) such that

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

(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:ℂP 1⟶Ω ∞−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)ℂP 1 ⟶Σ 21 E Ω ∞−2E ↓ ↗ c 1 E ℂP ∞ \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

*=ℂP 0↪ℂP 1↪ℂP 2↪ℂP 3↪⋯↪ℂP ∞=lim⟶ℂP • \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 ⟶ ℂP n+1 ↑ (po) ↑ S 2n+1 ⟶h ℂ 2n+1 ℂP 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 ℂ 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:

* ⟶ ℂP n+1 ↑ (hpo) ↑ S 2n+1 ⟶h ℂ 2n+1 ℂP 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

* ⟶ ℂP n+1 ⟶c 1 E,n+1 Ω ∞−2E ↑ (hpo) ↑ ↗ c 1 E,n S 2n+1 ⟶h ℂ 2n+1 ℂP 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 ℂ 2n+1 ℂP 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 ℂ 2n+1) *(c 1 E,n)∈E˜ 2(S 2n+1)≃E 2n−1 \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).

Example

(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.

Proposition

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 E⊗1,1⊗c 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 ] ].

Proof

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

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

H •(ℂP n,E •(*))⇒E •(ℂP 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 •(ℂP n,ℤ)≃ℤ[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,ℤ)≃ℤH^2(S^2, \mathbb{Z})\simeq \mathbb{Z}, and since similarly the ordinary homology of ℂP 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(ℂP 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 •(ℂP 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((ℂP n) 2,(ℂP 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 ℂP 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(ℂP 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 ∞ •,•≃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:

ℰ ∞ p,•≃{E •(*) ifp≤2nandeven 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 ℰ ∞ 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

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

In conclusion, this gives an isomorphism of graded rings

E •(ℂP n)≃⊕pℰ ∞ 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 •((ℂP ∞) 2n+2)=E •(ℂP n+1)→E •(ℂP n)=E •((ℂP ∞) 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 •(ℂP ∞) ≃E •(lim⟶ nℂP n) ≃lim⟵ nE •(ℂP 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..

Lemma

There is a continuous function

μ:ℂP ∞×ℂP⟶ℂP ∞ \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,ℂP ∞×ℂP ∞] ≃ ℂLineBund(X) /∼×ℂLineBund(X) /∼ [X,μ]↓ ↓ ⊗ [X,ℂP ∞] ≃ ℂLineBund(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 ℂLineBund(X) /∼\mathbb{C}LineBund(X)_{/\sim} denotes the set of isomorphism classes of complex line bundles on XX.

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

Proposition

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

E •(ℂP ∞)≃π •(E)[[c 1 E]],E •(ℂP ∞×ℂP ∞)≃π •(E)[[c 1 E⊗1,1⊗c 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 •(ℂP ∞)⟶E •(ℂP ∞×ℂP ∞)≃π •(E)[[c 1 E⊗1,1⊗c 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.

Proof

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. .

Proposition

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

(BU(1)) n⟶BU(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 n↪E •((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. ).

Proof

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(n−1)→BU(n)B U(n-1) \to B U(n).

This follows form the fact that S 2n−1≃U(n)/U(n−1)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(n−1)U(n-1)

U(n) ⟶ * ↓ BU(n)↦S 2n−1≃ U(n)/U(n−1) ⟶ BU(n−1) ↓ 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(n−1)→BU(n)B U(n-1) \to B U(n).

Proposition

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.

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:

• Douglas Ravenel, section 3 of: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)

• Michael Hopkins, Stable decompositions of certain loop spaces, Northwestern 1984 (pdf)

• Douglas Ravenel, section 6.5 of: Complex cobordism and stable homotopy groups of spheres, 1986

• Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Theorem 3 of: Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (jstor:1971440)

• Doug Ravenel, The first Adams-Novikov differential for the spectrum T(m)T(m), 2000 (pdf, pdf)

• Ippei Ichigi, Katsumi Shimomura, The Modulo Two Homotopy Groups of the L 2L_2-Localization of the Ravenel Spectrum, CUBO A Mathematical Journal, Vol. 10, No 03, (43–55). October 2008 (cubo:1498)

• Gabe Angelini-Knoll, J. D. Quigley: The Segal Conjecture for topological Hochschild homology of the Ravenel spectra, Journal of Topology 4 3 (2011) 591-622 [doi:10.1112/jtopol/jtr015, arXiv:1705.03343]

• Jonathan Beardsley, A Theorem on Multiplicative Cell Attachments with an Application to Ravenel’s X(n)X(n) Spectra, Journal of Homotopy and Related Structures volume 14, pages 611–624 (2019) (arXiv:1708.03042, doi:10.1007/s40062-018-0222-6)

• Jonathan Beardsley, Topological Hochschild homology of X(n)X(n) (arXiv:1708.09486)

• Xiangjun Wang, Zihong Yuan, The homotopy groups of L 2T(m)/(p [m2]+2,v 1)L_2 T(m)/\big(p^{[\tfrac{m}{2}]+2}, v_1 \big) for m>1m \gt 1, New York J. Math.24 (2018) 1123–1146 (pdf)