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cohomology

# Contents

## Statement

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

Hence $MU$ is the universal complex oriented cohomology theory.

## Details

##### Conner-Floyd-Chern classes are Thom classes

We discuss that for $E$ a complex oriented cohomology theory, then the $n$th universal Conner-Floyd-Chern class $c^E_n$ is in fact a universal Thom class for rank $n$ complex vector bundles. On the one hand this says that the choice of a complex orientation on $E$ 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 $M U \to E$ which represent that universal orienation.

###### Lemma

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

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

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

###### Proof

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

$(E U(n))/U(n-1) \longrightarrow (E U(n))/U(n)$

(by the proof of this prop.).

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

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

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

$E U(n) \underset{U(n)}{\times} \mathbb{C}^n \simeq \underset{\longrightarrow}{\lim}_k U(k)/U(k-n) \underset{U(n)}{\times} \mathbb{C}^n$

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

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

In particular:

###### Lemma

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

$B U(1) \overset{\in W_{cl}}{\longrightarrow} M U(1) \coloneqq Th( E U(1) \underset{U(1)}{\times} \mathbb{C}) \,.$
###### Proof

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

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

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

Therefore the Thom space

\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 $E$ a generalized (Eilenberg-Steenrod) cohomology theory, then the $E$-reduced cohomology of the Thom space of the complex universal vector bundle is equivalently the $E$-relative cohomology of $B U(n)$ relative $B U(n-1)$

$\tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) ) \;\simeq\; E^\bullet( B U(n), B U(n-1)) \,.$

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

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

where the $c_i$ are the universal $E$-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

\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 $E$-cohomology of $B U(n)$ for all $n$ (prop.):

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

and the restriction morphism

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

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

###### Proposition

Let $E$ be a complex oriented cohomology theory. Then the $n$th $E$-Conner-Floyd-Chern class

$c^E_n \in \tilde E(Th( E U(n) \underset{U(n)}{\times} \mathbb{C}^n ))$

(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 $\pi_0(E)$.

###### Proof

Since $B U(n)$ 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 $n$ complex lines, we may equivalently check that the restriction of $c^E_n$ to the pullback of the universal rank $n$ bundle along

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

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

$i^\ast c_n \simeq ( (c_1^E)_1 \cdots (c_1^E)_n ) \,.$

Hence it is now sufficient to see that each factor restricts to a unit on the fiber, but that it precisely the condition that $c_1^E$ is a complex orientaton of $E$. In fact by def. the restriction is even to $1 \in \pi_0(E)$.

##### Complex orientation as ring spectrum maps

For the present purpose:

###### Definition

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

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

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

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

is the unit

$\phi(c_1^E) = 1 \,.$
###### Remark

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

###### Lemma

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

$c_1^E \;\colon\; B U(1) \longrightarrow E_2 \,.$

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

$M U \longrightarrow E$

whose component map $M U_{2n} \longrightarrow E_{2n}$ represents $c_n^E$ (under the identification of lemma ), for all $n \in \mathbb{N}$.

###### Proof

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

$M U_{2n} \coloneqq Th( E U(n) \underset{}{\times} \mathbb{C}^n) \,.$

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

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

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

commute strictly (not just up to homotopy).

To begin with, pick a map

$f_0 \;\colon\; M U_0 \simeq S^0 \longrightarrow E_0$

that represents $c_0 = 1$.

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

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

where the maps denoted $c_k$ 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 $c_1^E$ is a complex orientation, while the homotopy in the bottom square exhibits the Whitney sum formula for Chern classes (prop.).

Now since $M U$ 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 $f_{2n+1} \colon M U_{2(n+1)} \longrightarrow E_{2(n+1)}$ homotopic to the given representative $c_{n+1}$ such that the required square commutes strictly.

###### Lemma

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

$c \;\colon\; M U \longrightarrow E$

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

###### Proof

The unitality condition demands that the diagram

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

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

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

commutes up to homotopy, hence that the restriction of $c_n$ to a fiber is the $2n$-fold suspension of the unit of $E_{2n}$. 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 $n \mapsto E^{n_1}(S^n)$ is actually constant (suspension isomorphism) hence trivially satisfies the Mittag-Leffler condition and so a homotopy of morphisms of spectra $\mathbb{S} \to E$ exists as soon as there are componentwise homotopies (cor.).

Next, the respect for the product demands that the square

$\array{ M U \wedge M U &\overset{c \wedge c}{\longrightarrow}& E \wedge E \\ \downarrow && \downarrow \\ M U &\underset{c}{\longrightarrow}& E }$

commutes in the stable homotopy category $Ho(Spectra)$. In order to rephrase this as a condition on the components of the ring spectra, regard this as happening in the homotopy category $Ho(OrthSpec(Top_{cg}))_{stable}$ 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

$\array{ M U_{2 n_1} \wedge M U_{2 n_2} &\overset{c_{n_1} \wedge c_{n_2}}{\longrightarrow}& E_{2 n_1} \wedge E_{2 n_2} \\ \downarrow && \downarrow \\ M U_{2(n_1 + n_2)} &\underset{c_{(n_1 + n_2)}}{\longrightarrow}& E_{2 (n_1 + n_2)} } \,,$

where on the left we have the standard pairing operations for $M U$ (def.) and on the right we have the given pairing on $E$.

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 $n \mapsto E^{n-1}((MU \wedge MU)_n)$-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 $E$ be a homotopy commutative ring spectrum (def.). Then the map

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

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

###### Proof

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

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

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

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

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

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

(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

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

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

## References

Last revised on May 12, 2017 at 05:47:06. See the history of this page for a list of all contributions to it.