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\begin{document}

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\section*{universal complex orientation on MU}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cobordism_theory}{}\paragraph*{{Cobordism theory}}\label{cobordism_theory}

[[!include cobordism theory -- contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{connerfloydchern_classes_are_thom_classes}{Conner-Floyd-Chern classes are Thom classes}\dotfill \pageref*{connerfloydchern_classes_are_thom_classes} \linebreak
\noindent\hyperlink{ComplexOrientationAsRingSpectrumMaps}{Complex orientation as ring spectrum maps}\dotfill \pageref*{ComplexOrientationAsRingSpectrumMaps} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

For $E$ a [[homotopy commutative ring spectrum]], there is a [[bijection]] between [[complex oriented cohomology theory|complex orientations]] on $E$ and homotopy ring spectrum homomorphism $MU \longrightarrow E$ from [[MU]].

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

(e.g \hyperlink{LurieLect6}{Lurie 10, lect. 6, theorem 8}, \hyperlink{Ravenel}{Ravenel, chapter 4, lemma 4.1.13})

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{connerfloydchern_classes_are_thom_classes}{}\paragraph*{{Conner-Floyd-Chern classes are Thom classes}}\label{connerfloydchern_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 oriented cohomology theory|complex orientation]] on $E$ indeed universally [[orientation in generalized cohomology|orients]] all [[complex vector bundles]]. On the other hand, we interpret this fact \hyperlink{ComplexOrientationAsRingSpectrumMaps}{below} as the [[unitality]] condition on a [[homomorphism]] of [[homotopy commutative ring spectra]] $M U \to E$ which represent that universal orienation.

\begin{lemma}
\label{SphereBundleBunminus1}\hypertarget{SphereBundleBunminus1}{}
For $n \in \mathbb{N}$, the [[fiber sequence]] (\href{classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces}{prop.})

\begin{displaymath}
\itexarray{
    S^{2n-1} &\longrightarrow& B U(n-1)
    \\
    && \downarrow
    \\
    && B U(n)
  }
\end{displaymath}
exhibits $B U(n-1)$ as the [[sphere bundle]] of the [[universal vector bundle|universal complex vector bundle]] over $B U(n)$.

\end{lemma}
\begin{proof}
When exhibited by a fibration, here the vertical morphism is equivalently the quotient map

\begin{displaymath}
(E U(n))/U(n-1) \longrightarrow (E U(n))/U(n)
\end{displaymath}
(by the proof of \href{classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces}{this prop.}).

Now the [[universal principal bundle]] $E U(n)$ is (\href{classifying+space#EOn}{def.}) equivalently the colimit

\begin{displaymath}
E U(n) \simeq \underset{\longrightarrow}{\lim}_k  U(k)/U(k-n)
  \,.
\end{displaymath}
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]]

\begin{displaymath}
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
\end{displaymath}
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 $E U(n) \underset{U(n)}{\times} \mathbb{C}^n$.

\end{proof}
In particular:

\begin{lemma}
\label{UniversalComplexLineBundleThomSpace}\hypertarget{UniversalComplexLineBundleThomSpace}{}
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 vector bundle|universal]] [[complex line bundle]] is a [[weak homotopy equivalence]]

\begin{displaymath}
B U(1) 
    \overset{\in W_{cl}}{\longrightarrow}
  M U(1)
   \coloneqq
  Th( E U(1) \underset{U(1)}{\times} \mathbb{C})
  \,.
\end{displaymath}
\end{lemma}
\begin{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{displaymath}
\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}
  \,.
\end{displaymath}
But the [[universal principal bundle]] is [[contractible topological space|contractible]]

\begin{displaymath}
E U(1) \overset{\in W_{cl}}{\longrightarrow} \ast
  \,.
\end{displaymath}
(Alternatively this is the special case of lemma \ref{SphereBundleBunminus1} for $n = 0$.)

Therefore the [[Thom space]]

\begin{displaymath}
\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}
  \,.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{UniversalComplexVectorBundleThomSpace}\hypertarget{UniversalComplexVectorBundleThomSpace}{}
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 $B U(n-1)$

\begin{displaymath}
\tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
    \;\simeq\;
  E^\bullet( B U(n), B U(n-1))
  \,.
\end{displaymath}
If $E$ is equipped with the structure of a [[complex oriented cohomology theory]] then

\begin{displaymath}
\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 ] ]
  \,,
\end{displaymath}
where the $c_i$ are the universal $E$-[[Conner-Floyd-Chern classes]].

\end{lemma}
\begin{proof}
Regarding the first statement:

In view of lemma \ref{SphereBundleBunminus1} and using that the disk bundle is homotopy equivalent to the base space we have

\begin{displaymath}
\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}
  \,.
\end{displaymath}
Regarding the second statement: the [[Conner-Floyd-Chern classes]] freely generate the $E$-cohomology of $B U(n)$ for all $n$ (\href{Conner-Floyd+Chern+class#ConnerFloyedClasses}{prop.}):

\begin{displaymath}
E^\bullet(B U(n))
  \simeq
  \pi_\bullet(E)[ [ c^E_1, \cdots, c^E_n ] ]
  \,.
\end{displaymath}
and the restriction morphism

\begin{displaymath}
E^\bullet(B U(n))
   \longrightarrow
  E^{\bullet}(B U(n-1))
\end{displaymath}
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.

\end{proof}
\begin{prop}
\label{ThomClassesCFClass}\hypertarget{ThomClassesCFClass}{}
Let $E$ be a [[complex oriented cohomology theory]]. Then the $n$th $E$-[[Conner-Floyd-Chern class]]

\begin{displaymath}
c^E_n \in \tilde E(Th( E U(n) \underset{U(n)}{\times} \mathbb{C}^n ))
\end{displaymath}
(using the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}) is a [[Thom class]] in that its restriction to the Thom space of any fiber is a suspension of a unit in $\pi_0(E)$.

\end{prop}
(\hyperlink{Lurie10}{Lurie 10, lecture 5, prop. 6})

\begin{proof}
Since $B U(n)$ is [[connected topological space|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

\begin{displaymath}
i \colon B U(1)^n \longrightarrow B U(n)
\end{displaymath}
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

\begin{displaymath}
i^\ast c_n \simeq ( (c_1^E)_1 \cdots (c_1^E)_n )
  \,.
\end{displaymath}
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. \ref{StrictComplexOrientation} the restriction is even to $1 \in \pi_0(E)$.

\end{proof}
\hypertarget{ComplexOrientationAsRingSpectrumMaps}{}\paragraph*{{Complex orientation as ring spectrum maps}}\label{ComplexOrientationAsRingSpectrumMaps}

For the present purpose:

\begin{defn}
\label{StrictComplexOrientation}\hypertarget{StrictComplexOrientation}{}
For $E$ a [[generalized (Eilenberg-Steenrod) cohomology]] theory, then a \emph{[[complex oriented cohomology theory|complex orientation]]} on $E$ is a choice of element

\begin{displaymath}
c_1^E \in E^2(B U(1))
\end{displaymath}
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

\begin{displaymath}
\phi
  \;\colon\;
  \tilde E^2( B U(1) )
    \longrightarrow
  \tilde E^2 (S^2)
    \simeq
  \pi_0(E)
\end{displaymath}
is the unit

\begin{displaymath}
\phi(c_1^E) = 1
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Lurie10}{Lurie 10, lecture 4, def. 2})

\begin{remark}
\label{}\hypertarget{}{}
Often one just requires that $\phi(c_1^E)$ is \emph{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).

\end{remark}
\begin{lemma}
\label{S2SpectrumMapFromComplexOrientation}\hypertarget{S2SpectrumMapFromComplexOrientation}{}
Let $E$ be a [[homotopy commutative ring spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum}{def.}) equipped with a [[complex oriented cohomology theory|complex orientation]] (def. \ref{StrictComplexOrientation}) represented by a map

\begin{displaymath}
c_1^E
  \;\colon\;
  B U(1)
    \longrightarrow
  E_2
  \,.
\end{displaymath}
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]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#SequentialTSpectra}{def.})

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

\end{lemma}
\begin{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]]

\begin{displaymath}
M U_{2n}
    \coloneqq
  Th( E U(n) \underset{U(n)}{\otimes} \mathbb{C}^n)
  \,.
\end{displaymath}
Notice that this is a [[CW-spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum}{def.}, \href{Thom+space#ThomSpaceCWStructure}{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 \ref{UniversalComplexVectorBundleThomSpace}) such that all the squares

\begin{displaymath}
\itexarray{
    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}
  }
\end{displaymath}
commute strictly (not just up to homotopy).

To begin with, pick a map

\begin{displaymath}
f_0 \;\colon\; M U_0 \simeq S^0 \longrightarrow E_0
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    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)}
  }
  \,,
\end{displaymath}
where the maps denoted $c_k$ are any representatives of the Chern classes of the same name, under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}. 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 (\href{Chern+class#WhitneySumChernClasses}{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.

\end{proof}
\begin{lemma}
\label{HRingSpectrumS2SpectrumMapFromComplexOrientation}\hypertarget{HRingSpectrumS2SpectrumMapFromComplexOrientation}{}
For $E$ a [[complex oriented cohomology theory|complex oriented]] [[homotopy commutative ring spectrum]], the morphism of spectra

\begin{displaymath}
c \;\colon\; M U \longrightarrow E
\end{displaymath}
that represents the complex orientation by lemma \ref{S2SpectrumMapFromComplexOrientation} is a [[homomorphism]] of [[homotopy commutative ring spectra]].

\end{lemma}
(\hyperlink{Lurie10}{Lurie 10, lecture 6, prop. 6})

\begin{proof}
The unitality condition demands that the diagram

\begin{displaymath}
\itexarray{
    \mathbb{S} 
      &\overset{}{\longrightarrow}&
    M U
    \\
    & \searrow & \downarrow^{\mathrlap{c}}
    \\
    && E
  }
\end{displaymath}
commutes in the [[stable homotopy category]] $Ho(Spectra)$. In components this means that

\begin{displaymath}
\itexarray{
    S^{2n} 
      &\overset{}{\longrightarrow}&
    M U_{2n}
    \\
    & \searrow & \downarrow^{\mathrlap{c_n}}
    \\
    && E_{2n}
  }
\end{displaymath}
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. \ref{ThomClassesCFClass}: 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 (\href{lim^1+and+Milnor+sequences#CohomologyOfSpectraMilnorSequence}{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 (\href{lim^1+and+Milnor+sequences#WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}{cor.}).

Next, the respect for the product demands that the square

\begin{displaymath}
\itexarray{
    M U \wedge M U
      &\overset{c \wedge c}{\longrightarrow}&
    E \wedge E
    \\
    \downarrow 
      && 
    \downarrow
    \\
    M U 
      &\underset{c}{\longrightarrow}&
    E
  }
\end{displaymath}
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 of a model category|homotopy category]] $Ho(OrthSpec(Top_{cg}))_{stable}$ of the [[model structure on orthogonal spectra]], which is [[equivalence of categories|equivalent]] to the [[stable homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SequentialSpectraQuillenEquivalence}{thm.}).

Here the derived [[symmetric monoidal smash product of spectra]] is given by [[Day convolution]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SsymModuleSymmetricSpectra}{def.}) and maps out of such a product are equivalently as in the above diagram is equivalent (\href{Introduction+to+Stable+homotopy+theory+--+1-2#DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor}{cor.}) to a suitably equivariant collection diagrams of the form

\begin{displaymath}
\itexarray{
    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)}
  }
  \,,
\end{displaymath}
where on the left we have the standard pairing operations for $M U$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalComplexThomSpectrum}{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 (\href{Chern+class#WhitneySumChernClasses}{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]] (\href{Conner-Floyd+Chern+class#ConnerFloyedClasses}{prop.}). Therefore these componentwise homotopies imply the required homotopy of morphisms of spectra (\href{lim^1+and+Milnor+sequences#WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}{cor.}).

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
Let $E$ be a [[homotopy commutative ring spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum}{def.}). Then the map

\begin{displaymath}
(M U \overset{c}{\longrightarrow} E)
  \;\mapsto\;
  (B U(1) \simeq M U_{2} \overset{c_1}{\longrightarrow} E_2)
\end{displaymath}
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 \ref{UniversalComplexLineBundleThomSpace}, constitutes a [[bijection]] from homotopy classes of homomorphisms of homotopy commutative ring spectra to complex orientations (def. \ref{StrictComplexOrientation}) on $E$.

\end{theorem}
(\hyperlink{Lurie10}{Lurie 10, lecture 6, theorem 8})

\begin{proof}
By lemma \ref{S2SpectrumMapFromComplexOrientation} and lemma \ref{HRingSpectrumS2SpectrumMapFromComplexOrientation} 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

\begin{displaymath}
f \;\colon\; M U \wedge I_+ \longrightarrow E 
  \,,
\end{displaymath}
This follows due to the existence of the [[Milnor exact sequence|Milnor]] [[short exact sequence]] of the form

\begin{displaymath}
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
\end{displaymath}
(\href{lim^1+and+Milnor+sequences#CohomologyOfSpectraMilnorSequence}{prop.}).

Here the [[Mittag-Leffler condition]] is clearly satisfied (by lemma \ref{UniversalComplexVectorBundleThomSpace} all relevant maps are epimorphisms). Hence the [[lim{\tt \symbol{94}}1]]-term vanishes, and so by exactness the canonical morphism

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

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Lazard's theorem]]


\item [[Landweber exact functor theorem]]


\item [[Quillen's theorem on MU]]


\item [[Landweber-Novikov theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Frank Adams]], lemma 4.6, example 4.7 in \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]},1986


\item [[Stanley Kochmann]], section 4.4 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, Lecture series 2010 (\href{http://www.math.harvard.edu/~lurie/252x.html}{web}), Lecture 6 \emph{MU and complex orientations} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf}{pdf})



\end{itemize}
[[!redirects MU and complex orientation]]

[[!redirects complex orientation and MU]] [[!redirects complex orientations and MU]]

[[!redirects universal complex orientation of MU]] [[!redirects MU and complex orientations]]



\end{document}