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\begin{document}

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\section*{orientation in generalized cohomology}

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

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

[[!include cohomology - contents]]

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{TraditionalDefinition}{Concretely}\dotfill \pageref*{TraditionalDefinition} \linebreak
\noindent\hyperlink{orientation_of_a_vector_bundle}{Orientation of a vector bundle}\dotfill \pageref*{orientation_of_a_vector_bundle} \linebreak
\noindent\hyperlink{orientation_of_a_manifold}{Orientation of a manifold}\dotfill \pageref*{orientation_of_a_manifold} \linebreak
\noindent\hyperlink{UniversalOrientationOfVectorBundles}{Universal orientation of vector bundles}\dotfill \pageref*{UniversalOrientationOfVectorBundles} \linebreak
\noindent\hyperlink{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{principal_bundles}{$GL_1(R)$-principal $\infty$-bundles}\dotfill \pageref*{principal_bundles} \linebreak
\noindent\hyperlink{associated_bundles}{$GL_1(R)$-associated $\infty$-bundles}\dotfill \pageref*{associated_bundles} \linebreak
\noindent\hyperlink{orientations}{$R$-Orientations}\dotfill \pageref*{orientations} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_between_thom_classes_and_trivializations}{Relation between Thom classes and trivializations}\dotfill \pageref*{relation_between_thom_classes_and_trivializations} \linebreak
\noindent\hyperlink{RelationToGenera}{Relation to genera}\dotfill \pageref*{RelationToGenera} \linebreak
\noindent\hyperlink{relation_to_cubical_structures}{Relation to cubical structures}\dotfill \pageref*{relation_to_cubical_structures} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{orientations_related_to_genera_and_indices}{Orientations related to genera and indices}\dotfill \pageref*{orientations_related_to_genera_and_indices} \linebreak
\noindent\hyperlink{complex_orientation}{Complex orientation}\dotfill \pageref*{complex_orientation} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Generally, for $E$ an [[E-∞ ring]] [[spectrum]], and $P \to X$ a [[sphere spectrum]]-bundle, an \emph{$E$-orientation} of $P$ is a trivialization of the [[associated bundle|associated]] $E$-bundle.

Specifically, for $P = Th(V)$ the [[Thom space]] of a [[vector bundle]] $V \to X$, an $E$-orientation of $V$ is an $E$-orientation of $P$.

More generally, for $A$ an $E$-algebra spectrum, an $E$-bundle is $A$-orientable if the associated $A$-bundle is trivializable. For more on this see [[(∞,1)-vector bundle]].

The existence of an $E$-orientation is necessary in order to have a notion of [[fiber integration]] in $E$-cohomology.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{TraditionalDefinition}{}\subsubsection*{{Concretely}}\label{TraditionalDefinition}

\hypertarget{orientation_of_a_vector_bundle}{}\paragraph*{{Orientation of a vector bundle}}\label{orientation_of_a_vector_bundle}

\begin{defn}
\label{EOrientationOfAVectorBundle}\hypertarget{EOrientationOfAVectorBundle}{}
Let $E$ be a [[multiplicative cohomology theory]] and let $V \to X$ be a topological [[vector bundle]] of [[rank]] $n$. Then an \textbf{$E$-orientation} or \textbf{$E$-[[Thom class]]} on $V$ is an element of degree $n$

\begin{displaymath}
u \in \tilde E^n(Th(V))
\end{displaymath}
in the [[reduced cohomology|reduced]] $E$-[[cohomology ring]] of the [[Thom space]] of $V$, such that for every point $x \in X$ its restriction $i_x^* u$ along

\begin{displaymath}
i_x 
    \;\colon\; 
  S^n \simeq Th(\mathbb{R}^n) \overset{Th(j_x)}{\longrightarrow} Th(V)
\end{displaymath}
(for $\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V$ the [[fiber]] of $V$ over $x$) is a \emph{generator}, in that it is of the form

\begin{displaymath}
i^\ast u = \epsilon \cdot \gamma_n
\end{displaymath}
for

\begin{itemize}%
\item $\epsilon \in \tilde E^0(S^0)$ a [[unit]] in $E^\bullet$;


\item $\gamma_n \in \tilde E^n(S^n)$ the image of the multiplicative unit under the [[suspension isomorphism]] $\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n)$.



\end{itemize}
\end{defn}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, def. 4.3.4})

\hypertarget{orientation_of_a_manifold}{}\paragraph*{{Orientation of a manifold}}\label{orientation_of_a_manifold}

\begin{defn}
\label{EOrientationOfAManifold}\hypertarget{EOrientationOfAManifold}{}
Let $E$ be a [[multiplicative cohomology theory]] and let $X$ be a [[manifold]], possibly [[manifold with boundary|with boundary]], of [[dimension]] $n$. An \textbf{$E$-orientation} of $X$ is a class in the $E$-[[generalized homology]]

\begin{displaymath}
\iota \in E_n(X,\partial X)
\end{displaymath}
with the property that for each point $x \in Int(X)$ in the [[interior]], it maps to a generator of $E_\bullet(\ast)$ under the map

\begin{displaymath}
E_\bullet(X,\partial X)
    \longrightarrow
  E_\bullet(X,\; X - \{x\})
    \simeq
  E_\bullet(D^n, S^{n-1})
    \simeq
  E_{\bullet-n}(\ast)
  \,,
\end{displaymath}
where the isomorphism is the excision isomorphism (\href{generalized+homology#excision}{def.}) for the complement of a closed [[n-ball]] around $x$.

\end{defn}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, p. 134})

\begin{prop}
\label{}\hypertarget{}{}
$E$-orientations of manifolds (def. \ref{EOrientationOfAManifold}) are equivalent to $E$-orientations of their stable [[normal bundle]] (def. \ref{EOrientationOfAVectorBundle}).

\end{prop}
(e.g. \hyperlink{Rudyak98}{Rudyak 98, chapter V, theorem 2.4}) (also \hyperlink{Kochmann96}{Kochmann 96, prop. 4.3.5}, but maybe that proof needs an extra argument)

\hypertarget{UniversalOrientationOfVectorBundles}{}\paragraph*{{Universal orientation of vector bundles}}\label{UniversalOrientationOfVectorBundles}

\begin{remark}
\label{}\hypertarget{}{}
Recall that a \emph{[[(B,f)-structure]]} $\mathcal{B}$ is a system of [[Serre fibrations]] $B_n \overset{f_n}{\longrightarrow} B O(n)$ over the [[classifying spaces]] for [[orthogonal structure]] equipped with maps

\begin{displaymath}
j_n \;\colon\; B_n \longrightarrow B_{n+1}
\end{displaymath}
covering the canonical inclusions of classifying spaces. For instance for $G_n \to O(n)$ a compatible system of [[topological group]] [[homomorphisms]], then the $(B,f)$-structure given by the [[classifying spaces]] $B G_n$ (possibly suitably resolved for the maps $B G_n \to B O(n)$ to become Serre fibrations) defines \emph{[[G-structure]]}.

Given a $(B,f)$-structure, then there are the [[pullbacks]] $V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$ of the [[universal vector bundles]] over $B O(n)$, which are the \emph{universal vector bundles equipped with $(B,f)$-structure}

\begin{displaymath}
\itexarray{
    V^{\mathcal{B}}_n &\longrightarrow& E O(n)\underset{O(n)}{\times} \mathbb{R}^n
    \\
    \downarrow &(pb)& \downarrow
    \\
    B_n & \underset{f_n}{\longrightarrow} & B O(n)
  }
  \,.
\end{displaymath}
Finally recall that there are canonical morphisms (\href{Thom+spectrum#PullbackOfUniversalOnBundleUnderCoordinateRestriction}{prop.})

\begin{displaymath}
\phi_n 
    \;\colon\; 
  \mathbb{R} \oplus V^{\mathcal{B}}_n 
    \longrightarrow 
  V^{\mathcal{B}}_{n+1}
\end{displaymath}
\end{remark}
\begin{defn}
\label{EOrientationOfABfStructure}\hypertarget{EOrientationOfABfStructure}{}
Let $E$ be a [[multiplicative cohomology theory]] and let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]]. Then a \textbf{universal $E$-orientation for vector bundles with $\mathcal{B}$-structure} is an $E$-orientation, according to def. \ref{EOrientationOfAVectorBundle}, for each rank-$n$ universal vector bundle with $\mathcal{B}$-structure:

\begin{displaymath}
\xi_n 
    \in 
  \tilde E^n(Th(V_n^{\mathcal{B}}))
  \;\;\;\;
  \forall n \in \mathbb{N}
\end{displaymath}
such that these are compatible in that

\begin{enumerate}%
\item for all $n \in \mathbb{N}$ then

\begin{displaymath}
\xi_n = \phi_n^\ast \xi_{n+1}
  \,,
\end{displaymath}
where

\begin{displaymath}
\xi_n 
   \in 
  \tilde E^n(Th(V_n)) 
    \simeq 
  \tilde E^{n+1}(\Sigma Th(V_n))
     \simeq 
  \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
\end{displaymath}
(with the first isomorphism is the [[suspension isomorphism]] of $E$ and the second exhibiting the [[homeomorphism]] of Thom spaces $Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V)$ (\href{Thom+space#SuspensionOfThomSpaces}{prop.})) and where

\begin{displaymath}
\phi_n^\ast
  \;\colon\;
   \tilde E^{n+1}(Th(V_{n+1}))
   \longrightarrow
  \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
\end{displaymath}
is pullback along the canonical $\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1}$ (\href{Thom+spectrum#PullbackOfUniversalOnBundleUnderCoordinateRestriction}{prop.}).


\item for all $n_1, n_2 \in \mathbb{N}$ then

\begin{displaymath}
\xi_{n+1} \cdot \xi_{n+2} = \xi_{n_1 + n_2}
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{prop}
\label{UniversalEOrientationsAreEquivalentlyMorphismsOfRingSpectra}\hypertarget{UniversalEOrientationsAreEquivalentlyMorphismsOfRingSpectra}{}
A universal $E$-orientation, in the sense of def. \ref{EOrientationOfABfStructure}, for vector bundles with [[(B,f)-structure]] $\mathcal{B}$, is equivalently (the homotopy class of) a homomorphism of [[ring spectra]]

\begin{displaymath}
\xi \;\colon\; M\mathcal{B} \longrightarrow E
\end{displaymath}
from the universal $\mathcal{B}$-[[Thom spectrum]] to a spectrum which via the [[Brown representability theorem]] represents the given [[generalized (Eilenberg-Steenrod) cohomology theory]] $E$ (and which we denote by the same symbol).

\end{prop}
\begin{proof}
The [[Thom spectrum]] $M\mathcal{B}$ has a standard structure of a [[CW-spectrum]]. Let now $E$ denote a [[sequential spectrum|sequential]] [[Omega-spectrum]] representing the multiplicative cohomology theory of the same name. Since, in the standard [[model structure on topological sequential spectra]], [[CW-spectra]] are cofibrant (\href{Introduction+to+Stable+homotopy+theory+--+1#CellSpectraAreCofibrantInModelStructureOnTopologicalSequentialSpectra}{prop.}) and Omega-spectra are fibrant (\href{Introduction+to+Stable+homotopy+theory+--+1#StableModelStructureOnSequentialSpectraIsModelCategory}{thm.}) we may represent all morphisms in the [[stable homotopy category]] (\href{Introduction+to+Stable+homotopy+theory+--+1#TheStableHomotopyCategory}{def.}) by actual morphisms

\begin{displaymath}
\xi
   \;\colon\;
  M \mathcal{B}
    \longrightarrow
  E
\end{displaymath}
of sequential spectra (due to \href{Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory}{this lemma}).

Now by definition (\href{Introduction+to+Stable+homotopy+theory+--+1#SequentialSpectra}{def.}) such a homomorphism is precissely a sequence of base-point preserving [[continuous functions]]

\begin{displaymath}
\xi_n 
    \;\colon\;
  (M\mathcal{B})_n
   =
  Th(V_n^{\mathcal{B}})
    \longrightarrow
  E_n
\end{displaymath}
for $n \in \mathbb{N}$, such that they are compatible with the structure maps $\sigma_n$ and equivalently with their $(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)$-[[adjuncts]] $\tilde \sigma_n$, in that these diagrams commute:

\begin{displaymath}
\itexarray{
    S^1 \wedge Th(V^{\mathcal{B}}_n)
     &\overset{S^1 \wedge \xi_n}{\longrightarrow}&
    S^1 \wedge E_n
    \\
    {}^{\mathllap{\sigma^{M\mathcal{B}}_n}}\downarrow 
      && 
    \downarrow^{\mathrlap{\sigma^E_n}}
    \\
    Th(V^{\mathcal{B}}_{n+1})
      &\underset{\xi_{n+1}}{\longrightarrow}&
    E_{n+1}
  }
  \;\;\;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;\;\;
  \itexarray{
     Th(V^{\mathcal{B}}_n)
       &\overset{\xi_n}{\longrightarrow}&
     E_n
     \\
     {}^{\mathllap{\tilde \sigma^{M\mathcal{B}}_n}}\downarrow 
       && 
     \downarrow^{\mathrlap{\tilde \sigma^E_n}}
     \\
     Maps(S^1,Th(V^{\mathcal{B}}_{n+1}))
       &\underset{Maps(S^1,\xi_{n+1})_\ast}{\longrightarrow}&
     Maps(S^1, E_{n+1})_{\ast}
  }
\end{displaymath}
for all $n \in \mathbb{N}$.

First of all this means (via the identification given by the [[Brown representability theorem]], see \href{Introduction+to+Stable+homotopy+theory+--+S#AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}{this prop.}) that the components $\xi_n$ are equivalently representatives of elements in the [[cohomology groups]]

\begin{displaymath}
\xi_n \in \tilde E^n(Th(V^{\mathcal{B}}_n))
\end{displaymath}
(which we denote by the same symbol, for brevity).

Now by the definition of universal [[Thom spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+S#UniversalThomSpectrum}{def.}, \href{Introduction+to+Stable+homotopy+theory+--+S#UniversalThomSpectrumForBfStructure}{def.}), the structure map $\sigma_n^{M\mathcal{B}}$ is just the map $\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}})$ from above.

Moreover, by the [[Brown representability theorem]], the [[adjunct]] $\tilde \sigma_n^E \circ \xi_n$ (on the right) of $\sigma^E_n \circ S^1 \wedge \xi_n$ (on the left) is what represents (again by \href{Introduction+to+Stable+homotopy+theory+--+S#AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum}{this prop.}) the image of

\begin{displaymath}
\xi_n \in E^n(Th(V^{\mathcal{B}}_n))
\end{displaymath}
under the [[suspension isomorphism]]. Hence the [[commutative square|commutativity]] of the above squares is equivalently the first compatibility condition from def. \ref{EOrientationOfABfStructure}: $\xi_n \simeq \phi_n^\ast \xi_{n+1}$ in $\tilde E^{n+1}(Th(\mathbb{R}\oplus V_n^{\mathcal{B}}))$

Next, $\xi$ being a homomorphism of [[ring spectra]] means equivalently (we should be modelling $M\mathcal{B}$ and $E$ as [[structured spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1#DiagramSpectra}{here.}) to be more precise on this point, but the conclusion is the same) that for all $n_1, n_2\in \mathbb{N}$ then

\begin{displaymath}
\itexarray{
    Th(V_{n_1}^{\mathcal{B}}) \wedge Th(V_{n_2}^{\mathcal{B}})
      &\overset{}{\longrightarrow}&
    Th(V_{n_1 + n_2})
    \\
    {}^{\mathllap{\xi_{n_1} \wedge \xi_{n_2}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\xi_{n_1 + n_2}}}
    \\
    E_{n_1} \wedge E_{n_2}
      &\underset{\cdot}{\longrightarrow}&
    E_{n_1 + n_2}
  }
  \,.
\end{displaymath}
This is equivalently the condition $\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1  + n_2}$.

Finally, since $M\mathcal{B}$ is a [[ring spectrum]], there is an essentially unique multiplicative homomorphism from the [[sphere spectrum]]

\begin{displaymath}
\mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
  \,.
\end{displaymath}
This is given by the component maps

\begin{displaymath}
e_n 
    \;\colon\;
  S^n 
    \simeq
  Th(\mathbb{R}^n)
    \longrightarrow
  Th(V_{n}^{\mathcal{B}})
\end{displaymath}
that are induced by including the fiber of $V_{n}^{\mathcal{B}}$.

Accordingly the composite

\begin{displaymath}
\mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
    \overset{\xi}{\longrightarrow}
  E
\end{displaymath}
has as components the restrictions $i^\ast \xi_n$ appearing in def. \ref{EOrientationOfAVectorBundle}. At the same time, also $E$ is a ring spectrum, hence it also has an essentially unique multiplicative morphism $\mathbb{S} \to E$, which hence must agree with $i^\ast \xi$, up to homotopy. If we represent $E$ as a [[symmetric ring spectrum]], then the canonical such has the required property: $e_0$ is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence $e_n$ is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.

\end{proof}
\hypertarget{abstractly}{}\subsubsection*{{Abstractly}}\label{abstractly}

Let $E$ be a [[E-∞ ring]] [[spectrum]]. Write $\mathbb{S}$ for the [[sphere spectrum]].

For the following see also May,Sigurdsson: \emph{[[Parametrized Homotopy Theory]]} (\href{http://mathoverflow.net/a/107837/381}{MO comment})

\hypertarget{principal_bundles}{}\paragraph*{{$GL_1(R)$-principal $\infty$-bundles}}\label{principal_bundles}

Write $R^\times$ or $GL_1(R)$ for the [[general linear group]] of the $E_\infty$-ring $R$: it is the subspace of the degree-0 space $\Omega^\infty R$ on those points that map to multiplicatively invertible elements in the ordinary ring $\pi_0(R)$.

Since $R$ is $E_\infty$, the space $GL_1(R)$ is itself an [[infinite loop space]]. Its one-fold [[delooping]] $B GL_1(R)$ is the [[classifying space]] for $GL_1(R)$-[[principal ∞-bundle]]s (in [[Top]]): for $X \in Top$ and $\zeta : X \to B GL_1(R)$ a map, its [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    GL_1(R) &\to& P &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{x}{\to}& X
   &\stackrel{\zeta}{\to}&
   B GL_1(R)
  }
\end{displaymath}
is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.

\begin{example}
\label{}\hypertarget{}{}
For $R = \mathbb{S}$ the [[sphere spectrum]], we have that $B GL_1(\mathbb{S})$ is the [[classifying space]] for spherical fibrations.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
There is a canonical morphism

\begin{displaymath}
B O \to B GL_1(\mathbb{S})
\end{displaymath}
from the classifying space of the [[orthogonal group]] to that of the [[infinity-group of units]] of the [[sphere spectrum]], called the \emph{[[J-homomorphism]]}. Postcomposition with this sends real [[vector bundle]]s $V \to X$ to sphere bundles. This is what is modeled by the [[Thom space]] construction

\begin{displaymath}
J : V \mapsto S^V
\end{displaymath}
which sends each fiber to its [[one-point compactification]].

\end{example}
\hypertarget{associated_bundles}{}\paragraph*{{$GL_1(R)$-associated $\infty$-bundles}}\label{associated_bundles}

For $P \to X$ a $GL_1(R)$-[[principal ∞-bundle]] there is canonically the corresponding [[associated ∞-bundle]] with fiber $R$. Precisely, in the [[stable (∞,1)-category]] $Stab(Top)$ of [[spectra]], regarded as the [[stabilization]] of the [[(∞,1)-topos]] [[Top]]

\begin{displaymath}
Stab(Top) \simeq Spectra
   \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}}
  Top
\end{displaymath}
the associated bundle is the [[smash product]] over $\Sigma^\infty GL_1(R)$

\begin{displaymath}
X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R
  \,.
\end{displaymath}
This is the generalized \textbf{[[Thom spectrum]]}. For $R = K O$ the real [[K-theory spectrum]] this is given by the ordinary [[Thom space]] construction on a [[vector bundle]] $V \to X$.

An $E$-orientation of a vector bundle $V \to X$ is a trivialization of the $E$-module bundle $E \wedge S^V$, where we fiberwise form the [[smash product]] of $E$ with the [[Thom space]] of $V$.

\begin{prop}
\label{}\hypertarget{}{}
For $f : R \to S$ a morphism of $E_\infty$-rings, and $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite

\begin{displaymath}
X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)
\end{displaymath}
is given by the [[smash product]]

\begin{displaymath}
X^{f \circ \zeta} \simeq X^\zeta \wedge_R S
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Hopkins}{Hopkins, bottom of p. 6}).

\hypertarget{orientations}{}\paragraph*{{$R$-Orientations}}\label{orientations}

For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an \textbf{$R$-orientation} on $X^\zeta$ is a trivialization of the associated $R$-bundle $X^\zeta \wedge R$, hence a trivialization (null-homotopy) of the classifying morphism

\begin{displaymath}
X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
  \,,
\end{displaymath}
where the second map comes from the unit of $E_\infty$-rings $\mathbb{S} \to R$ (the [[sphere spectrum]] is the [[initial object]] in $E_\infty$-rings).

Specifically, for $V : X \to B O$ a [[vector bundle]], an $E$-orientation on it is a trivialization of the $R$-bundle associated to the associated [[Thom space]] sphere bundle, hence a trivialization of the morphism

\begin{displaymath}
\itexarray{
  B O &\stackrel{J}{\to}& B GL_1(\mathbb{S})
    &\stackrel{\iota}{\to}&
   B GL_1(R)
   \\
   {}^{\mathllap{V}}\uparrow & \nearrow_{\mathrlap{\zeta}}
   \\
   X
  }
  \,.
\end{displaymath}
This appears as (\hyperlink{Hopkins}{Hopkins, p.7}).

A natural $R$-orientation of \emph{all} vector bundles is therefore a trivialization of the morphism

\begin{displaymath}
B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
 \,.
\end{displaymath}
Similarly, an $R$-orientation of \emph{all} [[spinor bundle]]s is a trivialization of

\begin{displaymath}
B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
\end{displaymath}
and an $R$-orientation of all [[string group]]-bundles a trivialization of

\begin{displaymath}
B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
\end{displaymath}
and so forth, through the [[Whitehead tower]] of $B O$.

Now, the [[Thom spectrum]] [[MO]] is the [[spherical fibration]] over $B O$ [[associated infinity-bundle|associated]] to the $O$-[[universal principal bundle]]. In generalization of the way that a trivialization of an ordinary $G$-principal bundle $P$ is given by a $G$-equivariant map $P \to G$, one finds that trivializations of the morphism

\begin{displaymath}
B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
\end{displaymath}
correspond to $E_\infty$-maps

\begin{displaymath}
M O \to R
\end{displaymath}
from the [[Thom spectrum]] to $R$. Similarly trivialization of

\begin{displaymath}
B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
\end{displaymath}
corresponds to morphisms

\begin{displaymath}
M Spin \to R
\end{displaymath}
and trivializations of

\begin{displaymath}
B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
\end{displaymath}
to morphisms

\begin{displaymath}
M String \to R
\end{displaymath}
and so forth.

This is the way orientations in generalized cohomology often appear in the literature.

\begin{example}
\label{}\hypertarget{}{}
The construction of the [[string orientation of tmf]], hence a morphism

\begin{displaymath}
M String \to tmf
\end{displaymath}
is discussed in (\hyperlink{Hopkins}{Hopkins, last pages}).

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_between_thom_classes_and_trivializations}{}\subsubsection*{{Relation between Thom classes and trivializations}}\label{relation_between_thom_classes_and_trivializations}

The relation (equivalence) between choices of [[Thom classes]] and trivializations of [[(∞,1)-line bundles]] is discussed e.g. in \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, section 3.3}

\hypertarget{RelationToGenera}{}\subsubsection*{{Relation to genera}}\label{RelationToGenera}

Let $G$ be a [[topological group]] equipped with a [[homomorphism]] to the [[stable orthogonal group]], and write $B G \to B O$ for the corresponding map of [[classifying spaces]]. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the [[J-homomorphism]].

For $E$ an [[E-∞ ring]], there is a canonical homomorphism $B GL_1(\mathbb{S}) \to B GL_1(E)$ between the [[deloopings]] of the [[∞-groups of units]]. A trivialization of the total composite

\begin{displaymath}
B G \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to B GL_1(E)
\end{displaymath}
is a universal $E$-orientation of [[G-structures]]. Under [[(∞,1)-colimit]] in $E Mod$ this induces a homomorphism of $E$-[[∞-modules]]

\begin{displaymath}
\sigma \;\colon\; M G \to E
\end{displaymath}
from the universal $G$-[[Thom spectrum]] to $E$.

If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of [[spectra]] then this $\sigma$ is a homomorphism of [[E-∞ rings]] and in this case there is a [[bijection]] between universal orientations and such $E_\infty$-ring homomorphisms (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 2.11}).

The latter, on passing to [[homotopy groups]], are [[genera]] on manifolds with [[G-structure]].

\hypertarget{relation_to_cubical_structures}{}\subsubsection*{{Relation to cubical structures}}\label{relation_to_cubical_structures}

For $E$ a [[multiplicative cohomology theory|multiplicative]] [[weakly periodic cohomology theory|weakly periodic]] [[complex orientable cohomology theory]] then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of [[cubical structure on a line bundle|cubical structures]] on the trivial line bundle over the [[formal group]] of $E$.

In particular, [[homotopy classes]] of maps of [[E-infinity ring]] spectra $MU\angle 6\rangle \to E$ from the [[Thom spectrum]] to $E$, and hence universal $MU\langle 6\rangle$-[[orientation in generalized cohomology|orientations]] (see there) of $E$ are in natural bijection with these cubical structures.

See at \emph{[[cubical structure]]} for more details and references. This way for instance the [[string orientation of tmf]] has been constructed. See there for more.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{orientations_related_to_genera_and_indices}{}\subsubsection*{{Orientations related to genera and indices}}\label{orientations_related_to_genera_and_indices}

[[!include genera and partition functions - table]]

\hypertarget{complex_orientation}{}\subsubsection*{{Complex orientation}}\label{complex_orientation}

An $E_\infty$ [[complex oriented cohomology theory]] $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see \href{complex+oriented+cohomology+theory#InTermsOfGenera}{here}.

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

\begin{itemize}%
\item [[differential Thom class]]


\item [[differential orientation]], [[fiber integration in differential cohomology]]


\item [[(infinity,1)-vector bundle]]



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

Discussion in terms of [[Thom classes]]:

\begin{itemize}%
\item [[Frank Adams]], part III, section 10 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Stanley Kochmann]], section 4.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Yuli Rudyak]], \emph{In Thom spectra, Orientability and Cobordism}, Springer 1998 (\href{http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf}{pdf})


\item [[Jacob Lurie]], lecture 5 of \emph{[[Chromatic Homotopy Theory]]}, 2010 (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf}{pdf})


\item [[eom]], \emph{\href{http://eom.springer.de/o/o070200.htm}{Orientation}}


\item [[Manifold Atlas]], \emph{\href{http://www.map.mpim-bonn.mpg.de/Orientation_of_manifolds_in_generalized_cohomology_theories}{Orientation of manifolds in generalized cohomology theories}}



\end{itemize}
A comprehensive account of the general abstract persepctive (with predecessors in \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10}) is in

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], \emph{Units of ring spectra and Thom spectra} (\href{http://arxiv.org/abs/0810.4535}{arXiv:0810.4535})


\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], section 6 of \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004})



\end{itemize}
Lecture notes on this include

\begin{itemize}%
\item [[Mike Hopkins]] (notes by [[André Henriques]]), \emph{The String orientation of tmf} (\href{http://arxiv.org/abs/0805.0743}{arXiv:0805.0743})

\end{itemize}
which are motivated towards constructing the [[string orientation of tmf]], based on

\begin{itemize}%
\item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})

\end{itemize}
[[!redirects Thom class]] [[!redirects Thom classes]]

[[!redirects orientations in generalized cohomology]]

[[!redirects E-orientation]] [[!redirects E-orientations]]



\end{document}