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universal complex orientation on MU

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Statement

For EE a homotopy commutative ring spectrum, there is a bijection between complex orientations on EE and homotopy ring spectrum homomorphism MUEMU \longrightarrow E from MU.

Hence MUMU is the universal complex oriented cohomology theory.

(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)

Details

Conner-Floyd-Chern classes are Thom classes

We discuss that for EE a complex oriented cohomology theory, then the nnth universal Conner-Floyd-Chern class c n Ec^E_n is in fact a universal Thom class for rank nn complex vector bundles. On the one hand this says that the choice of a complex orientation on EE indeed universally orients all complex vector bundles. On the other hand, we interpret this fact below as the unitality condition on a homomorphism of homotopy commutative ring spectra MUEM U \to E which represent that universal orienation.

Lemma

For nn \in \mathbb{N}, the fiber sequence (prop.)

S 2n1 BU(n1) BU(n) \array{ S^{2n-1} &\longrightarrow& B U(n-1) \\ && \downarrow \\ && B U(n) }

exhibits BU(n1)B U(n-1) as the sphere bundle of the universal complex vector bundle over BU(n)B U(n).

Proof

When exhibited by a fibration, here the vertical morphism is equivalently the quotient map

(EU(n))/U(n1)(EU(n))/U(n) (E U(n))/U(n-1) \longrightarrow (E U(n))/U(n)

(by the proof of this prop.).

Now the universal principal bundle EU(n)E U(n) is (def.) equivalently the colimit

EU(n)lim kU(k)/U(kn). E U(n) \simeq \underset{\longrightarrow}{\lim}_k U(k)/U(k-n) \,.

Here each Stiefel manifold/coset spaces U(k)/U(kn)U(k)/U(k-n) is equivalently the space of (complex) nn-dimensional subspaces of k\mathbb{C}^k that are equipped with an orthonormal (hermitian) linear basis. The universal vector bundle

EU(n)×U(n) nlim kU(k)/U(kn)×U(n) n E U(n) \underset{U(n)}{\times} \mathbb{C}^n \simeq \underset{\longrightarrow}{\lim}_k U(k)/U(k-n) \underset{U(n)}{\times} \mathbb{C}^n

has as fiber precisely the linear span of any such choice of basis.

While the quotient U(k)/(U(nk)×U(n))U(k)/(U(n-k)\times U(n)) (the Grassmannian) divides out the entire choice of basis, the quotient U(k)/(U(nk)×U(n1))U(k)/(U(n-k) \times U(n-1)) leaves the choice of precisly one unit vector. This is parameterized by the sphere S 2n1S^{2n-1} which is thereby identified as the unit sphere in the respective fiber of EU(n)×U(n) nE U(n) \underset{U(n)}{\times} \mathbb{C}^n.

In particular:

Lemma

The canonical map from the classifying space BU(1)P B U(1) \simeq \mathbb{C}P^\infty (the inifnity complex projective space) to the Thom space of the universal complex line bundle is a weak homotopy equivalence

BU(1)W clMU(1)Th(EU(1)×U(1)). B U(1) \overset{\in W_{cl}}{\longrightarrow} M U(1) \coloneqq Th( E U(1) \underset{U(1)}{\times} \mathbb{C}) \,.
Proof

Observe that the circle group U(1)U(1) is naturally identified with the unit sphere in \mathbb{C}: U(1)S(𝕊)U (1) \simeq S(\mathbb{S}). Therefore the sphere bundle of the universal complex line bundle is equivalently the U(1)U(1)-universal principal bundle

EU(1)×U(1)S() EU(1)×U(1)U(1) EU(1). \begin{aligned} E U(1) \underset{U(1)}{\times} S(\mathbb{C}) & \simeq E U(1) \underset{U(1)}{\times} U(1) \\ & \simeq E U(1) \end{aligned} \,.

But the universal principal bundle is contractible

EU(1)W cl*. E U(1) \overset{\in W_{cl}}{\longrightarrow} \ast \,.

(Alternatively this is the special case of lemma for n=0n = 0.)

Therefore the Thom space

Th(EU(1)×U(1)𝔹) D(EU(1)×U(1)𝔹)/S(EU(1)×U(1)𝔹) W clD(EU(1)×U(1)𝔹) W clBU(1). \begin{aligned} Th( E U(1) \underset{U(1)}{\times} \mathbb{B} ) & \coloneqq D( E U(1) \underset{U(1)}{\times} \mathbb{B} ) / S( E U(1) \underset{U(1)}{\times} \mathbb{B} ) \\ & \overset{\in W_{cl}}{\longrightarrow} D( E U(1) \underset{U(1)}{\times} \mathbb{B} ) \\ & \overset{\in W_{cl}}{\longrightarrow} B U(1) \end{aligned} \,.
Lemma

For EE a generalized (Eilenberg-Steenrod) cohomology theory, then the EE-reduced cohomology of the Thom space of the complex universal vector bundle is equivalently the EE-relative cohomology of BU(n)B U(n) relative BU(n1)B U(n-1)

E˜ (Th(EU(n)×U(n) n))E (BU(n),BU(n1)). \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) ) \;\simeq\; E^\bullet( B U(n), B U(n-1)) \,.

If EE is equipped with the structure of a complex oriented cohomology theory then

E˜ (Th(EU(n)×U(n) n))c n E(π (E))[[c 1 E,,c n E]], \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) ) \simeq c^E_n \cdot (\pi_\bullet(E))[ [ c^E_1, \cdots, c^E_n ] ] \,,

where the c ic_i are the universal EE-Conner-Floyd-Chern classes.

Proof

Regarding the first statement:

In view of lemma and using that the disk bundle is homotopy equivalent to the base space we have

E˜ (Th(EU(n)×U(n) n)) =E (D(EU(n)×U(n) n),S(EU(n)×U(n) n)) E (EU(n),BU(n1)). \begin{aligned} \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) ) & = E^\bullet( D(E U(n) \underset{U(n)}{\times} \mathbb{C}^n), S(E U(n) \underset{U(n)}{\times} \mathbb{C}^n) ) \\ & \simeq E^\bullet( E U(n), B U(n-1)) \end{aligned} \,.

Regarding the second statement: the Conner-Floyd-Chern classes freely generate the EE-cohomology of BU(n)B U(n) for all nn (prop.):

E (BU(n))π (E)[[c 1 E,,c n E]]. E^\bullet(B U(n)) \simeq \pi_\bullet(E)[ [ c^E_1, \cdots, c^E_n ] ] \,.

and the restriction morphism

E (BU(n))E (BU(n1)) E^\bullet(B U(n)) \longrightarrow E^{\bullet}(B U(n-1))

projects out c n Ec_n^E. Since this is in particular a surjective map, the relative cohomology E (BU(n),BU(n1))E^\bullet( B U(n), B U(n-1) ) is just the kernel of this map.

Proposition

Let EE be a complex oriented cohomology theory. Then the nnth EE-Conner-Floyd-Chern class

c n EE˜(Th(EU(n)×U(n) n)) c^E_n \in \tilde E(Th( E U(n) \underset{U(n)}{\times} \mathbb{C}^n ))

(using the identification of lemma ) is a Thom class in that its restriction to the Thom space of any fiber is a suspension of a unit in π 0(E)\pi_0(E).

(Lurie 10, lecture 5, prop. 6)

Proof

Since BU(n)B U(n) is connected, it is sufficient to check the statement over the base point. Since that fixed fiber is canonically isomorphic to the direct sum of nn complex lines, we may equivalently check that the restriction of c n Ec^E_n to the pullback of the universal rank nn bundle along

i:BU(1) nBU(n) i \colon B U(1)^n \longrightarrow B U(n)

satisfies the required condition. By the splitting principle, that restriction is the product of the nn-copies of the first EE-Conner-Floyd-Chern class

i *c n((c 1 E) 1(c 1 E) n). i^\ast c_n \simeq ( (c_1^E)_1 \cdots (c_1^E)_n ) \,.

Hence it is now sufficient to see that each factor restricts to a unit on the fiber, but that it precisely the condition that c 1 Ec_1^E is a complex orientaton of EE. In fact by def. the restriction is even to 1π 0(E)1 \in \pi_0(E).

Complex orientation as ring spectrum maps

For the present purpose:

Definition

For EE a generalized (Eilenberg-Steenrod) cohomology theory, then a complex orientation on EE is a choice of element

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

in the cohomology of the classifying space BU(1)B U(1) (given by the infinite complex projective space) such that its image under the restriction map

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

is the unit

ϕ(c 1 E)=1. \phi(c_1^E) = 1 \,.

(Lurie 10, lecture 4, def. 2)

Remark

Often one just requires that ϕ(c 1 E)\phi(c_1^E) is a unit, i.e. an invertible element. However we are after identifying c 1 Ec_1^E with the degree-2 component MU(1)E 2M U(1) \to E_2 of homtopy ring spectrum morphisms MUEM U \to E, and by unitality these necessarily send S 2MU(1)S^2 \to M U(1) to the unit ι 2:S 2E\iota_2 \;\colon\; S^2 \to E (up to homotopy).

Lemma

Let EE be a homotopy commutative ring spectrum (def.) equipped with a complex orientation (def. ) represented by a map

c 1 E:BU(1)E 2. c_1^E \;\colon\; B U(1) \longrightarrow E_2 \,.

Write {c k E} k\{c^E_k\}_{k \in \mathbb{N}} for the induced Conner-Floyd-Chern classes. Then there exists a morphism of S 2S^2-sequential spectra (def.)

MUE M U \longrightarrow E

whose component map MU 2nE 2nM U_{2n} \longrightarrow E_{2n} represents c n Ec_n^E (under the identification of lemma ), for all nn \in \mathbb{N}.

Proof

Consider the standard model of MU as a sequential S 2S^2-spectrum with component spaces the Thom spaces of the complex universal vector bundle

MU 2nTh(EU(n)U(n) n). M U_{2n} \coloneqq Th( E U(n) \underset{U(n)}{\otimes} \mathbb{C}^n) \,.

Notice that this is a CW-spectrum (def., lemma).

In order to get a homomorphism of S 2S^2-sequential spectra, we need to find representatives f 2n:MU 2nE 2nf _{2n} \;\colon\; M U_{2n} \longrightarrow E_{2n} of c n Ec^E_n (under the identification of lemma ) such that all the squares

S 2MU 2n idf 2n S 2E 2n MU 2(n+1) f 2(n+1) E 2n+1 \array{ S^2 \wedge M U_{2n} &\overset{id \wedge f_{2n}}{\longrightarrow}& S^2 \wedge E_{2n} \\ \downarrow && \downarrow \\ M U_{2(n+1)} &\underset{f_{2(n+1)}}{\longrightarrow}& E_{2n+1} }

commute strictly (not just up to homotopy).

To begin with, pick a map

f 0:MU 0S 0E 0 f_0 \;\colon\; M U_0 \simeq S^0 \longrightarrow E_0

that represents c 0=1c_0 = 1.

Assume then by induction that maps f 2kf_{2k} have been found for knk \leq n. Observe that we have a homotopy-commuting diagram of the form

S 2MU 2n idf 2n S 2E 2n MU 2MU 2n c 1c n E 2E 2n μ 2,2n MU 2(n+1) c n+1 E 2(n+1), \array{ S^2 \wedge M U_{2n} &\overset{id \wedge f_{2n}}{\longrightarrow}& S^2 \wedge E_{2n} \\ \downarrow &\swArrow& \downarrow \\ M U_{2} \wedge M U_{2 n} &\overset{c_1 \wedge c_{n}}{\longrightarrow}& E_2 \wedge E_{2n} \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\mu_{2,2n}}} \\ M U_{2(n+1)} &\underset{c_{n+1}}{\longrightarrow}& E_{2(n+1)} } \,,

where the maps denoted c kc_k are any representatives of the Chern classes of the same name, under the identification of lemma . Here the homotopy in the top square exhibits the fact that c 1 Ec_1^E is a complex orientation, while the homotopy in the bottom square exhibits the Whitney sum formula for Chern classes (prop.).

Now since MUM U is a CW-spectrum, the total left vertical morphism here is a (Serre-)cofibration, hence a Hurewicz cofibration, hence satisfies the homotopy extension property. This means precisely that we may find a map f 2n+1:MU 2(n+1)E 2(n+1)f_{2n+1} \colon M U_{2(n+1)} \longrightarrow E_{2(n+1)} homotopic to the given representative c n+1c_{n+1} such that the required square commutes strictly.

Lemma

For EE a complex oriented homotopy commutative ring spectrum, the morphism of spectra

c:MUE c \;\colon\; M U \longrightarrow E

that represents the complex orientation by lemma is a homomorphism of homotopy commutative ring spectra.

(Lurie 10, lecture 6, prop. 6)

Proof

The unitality condition demands that the diagram

𝕊 MU c E \array{ \mathbb{S} &\overset{}{\longrightarrow}& M U \\ & \searrow & \downarrow^{\mathrlap{c}} \\ && E }

commutes in the stable homotopy category Ho(Spectra)Ho(Spectra). In components this means that

S 2n MU 2n c n E 2n \array{ S^{2n} &\overset{}{\longrightarrow}& M U_{2n} \\ & \searrow & \downarrow^{\mathrlap{c_n}} \\ && E_{2n} }

commutes up to homotopy, hence that the restriction of c nc_n to a fiber is the 2n2n-fold suspension of the unit of E 2nE_{2n}. But this is the statement of prop. : the Chern classes are universal Thom classes.

Hence componentwise all these triangles commute up to some homotopy. Now we invoke the Milnor sequence for generalized cohomology of spectra (prop.) Observe that the tower of abelian groups nE n 1(S n)n \mapsto E^{n_1}(S^n) is actually constant (suspension isomorphism) hence trivially satisfies the Mittag-Leffler condition and so a homotopy of morphisms of spectra 𝕊E\mathbb{S} \to E exists as soon as there are componentwise homotopies (cor.).

Next, the respect for the product demands that the square

MUMU cc EE MU c E \array{ M U \wedge M U &\overset{c \wedge c}{\longrightarrow}& E \wedge E \\ \downarrow && \downarrow \\ M U &\underset{c}{\longrightarrow}& E }

commutes in the stable homotopy category Ho(Spectra)Ho(Spectra). In order to rephrase this as a condition on the components of the ring spectra, regard this as happening in the homotopy category Ho(OrthSpec(Top cg)) stableHo(OrthSpec(Top_{cg}))_{stable} of the model structure on orthogonal spectra, which is equivalent to the stable homotopy category (thm.).

Here the derived symmetric monoidal smash product of spectra is given by Day convolution (def.) and maps out of such a product are equivalently as in the above diagram is equivalent (cor.) to a suitably equivariant collection diagrams of the form

MU 2n 1MU 2n 2 c n 1c n 2 E 2n 1E 2n 2 MU 2(n 1+n 2) c (n 1+n 2) E 2(n 1+n 2), \array{ M U_{2 n_1} \wedge M U_{2 n_2} &\overset{c_{n_1} \wedge c_{n_2}}{\longrightarrow}& E_{2 n_1} \wedge E_{2 n_2} \\ \downarrow && \downarrow \\ M U_{2(n_1 + n_2)} &\underset{c_{(n_1 + n_2)}}{\longrightarrow}& E_{2 (n_1 + n_2)} } \,,

where on the left we have the standard pairing operations for MUM U (def.) and on the right we have the given pairing on EE.

That this indeed commutes up to homotopy is the Whitney sum formula for Chern classes (prop.).

Hence again we have componentwise homotopies. And again the relevant Mittag-Leffler condition on nE n1((MUMU) n)n \mapsto E^{n-1}((MU \wedge MU)_n)-holds, by the nature of the universal Conner-Floyd classes (prop.). Therefore these componentwise homotopies imply the required homotopy of morphisms of spectra (cor.).

Theorem

Let EE be a homotopy commutative ring spectrum (def.). Then the map

(MUcE)(BU(1)MU 2c 1E 2) (M U \overset{c}{\longrightarrow} E) \;\mapsto\; (B U(1) \simeq M U_{2} \overset{c_1}{\longrightarrow} E_2)

which sends a homomorphism cc of homotopy commutative ring spectra to its component map in degree 2, interpreted as a class on BU(1)B U(1) via lemma , constitutes a bijection from homotopy classes of homomorphisms of homotopy commutative ring spectra to complex orientations (def. ) on EE.

(Lurie 10, lecture 6, theorem 8)

Proof

By lemma and lemma the map is surjective, hence it only remains to show that it is injective.

So let c,c:MUEc, c' \colon M U \to E be two morphisms of homotopy commutative ring spectra that have the same restriction, up to homotopy, to c 1c 1:MU 2BU(1)c_1 \simeq c_1'\colon M U_2 \simeq B U(1). Since both are homotopy ring spectrum homomophisms, the restriction of their components c n,c n:MU 2nE 2nc_n, c'_n \colon M U_{2n} \to E_{2 n} to BU(1) nB U(1)^{\wedge^n} is a product of c 1c 1c_1 \simeq c'_1, hence c nc_n becomes homotopic to c nc_n' after this restriction. But by the splitting principle this restriction is injective on cohomology classes, hence c nc_n itself ist already homotopic to c nc'_n.

It remains to see that these homotopies may be chosen compatibly such as to form a single homotopy of maps of spectra

f:MUI +E, f \;\colon\; M U \wedge I_+ \longrightarrow E \,,

This follows due to the existence of the Milnor short exact sequence of the form

0lim n 1E 1(Σ 2nMU 2n)E 0(MU)lim nE 0(Σ 2nMU 2n)0 0 \to \underset{\longleftarrow}{\lim}^1_n E^{-1}( \Sigma^{-2n} M U_{2n} ) \longrightarrow E^0(M U) \longrightarrow \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} ) \to 0

(prop.).

Here the Mittag-Leffler condition is clearly satisfied (by lemma all relevant maps are epimorphisms). Hence the lim^1-term vanishes, and so by exactness the canonical morphism

E 0(MU)lim nE 0(Σ 2nMU 2n) E^0(M U) \overset{\simeq}{\longrightarrow} \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} )

is an isomorphism. This says that two homotopy classes of morphisms MUEM U \to E are equal precisely already if all their component morphisms are homotopic (represent the same cohomology class).

References

Last revised on April 26, 2019 at 05:07:40. See the history of this page for a list of all contributions to it.