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\begin{document}

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\section*{Thom spectrum}

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

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

[[!include cobordism theory -- contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy 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{ForVectorBundles}{For vector bundles}\dotfill \pageref*{ForVectorBundles} \linebreak
\noindent\hyperlink{ForSphereBundles}{For spherical fibrations}\dotfill \pageref*{ForSphereBundles} \linebreak
\noindent\hyperlink{ForInfinityModuleBundles}{For $(\infty,1)$-module bundles}\dotfill \pageref*{ForInfinityModuleBundles} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RingSpectrumStructure}{Ring spectrum structure}\dotfill \pageref*{RingSpectrumStructure} \linebreak
\noindent\hyperlink{relation_to_the_cobordism_ring}{Relation to the cobordism ring}\dotfill \pageref*{relation_to_the_cobordism_ring} \linebreak
\noindent\hyperlink{AsDualObject}{As a dual in the stable homotopy category}\dotfill \pageref*{AsDualObject} \linebreak
\noindent\hyperlink{AsTheUniversalSphericalFibration}{As the universal spherical fibration, from the $J$-homomorphism}\dotfill \pageref*{AsTheUniversalSphericalFibration} \linebreak
\noindent\hyperlink{AsTheInfiniteCobordismCategory}{As the infinite cobordism category}\dotfill \pageref*{AsTheInfiniteCobordismCategory} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{as_dual_objects_in_the_stable_homotopy_category}{As dual objects in the stable homotopy category}\dotfill \pageref*{as_dual_objects_in_the_stable_homotopy_category} \linebreak


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

The universal real Thom spectrum [[MO]] is a [[connective spectrum]] whose associated [[infinite loop space]] is the [[classifying space]] for [[cobordism]]:

\begin{displaymath}
\Omega^\infty M O \simeq \vert Cob_\infty \vert .
\end{displaymath}
In particular, $\pi_n M O$ is naturally identified with the set of [[cobordism classes]] of [[closed manifold|closed]] $n$-[[manifolds]] ([[Thom's theorem]]).

More abstractly, [[MO]] is the [[homotopy colimit]] of the [[J-homomorphism]] in [[Spectra]]

\begin{displaymath}
M O \simeq \underset{\longrightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)
\end{displaymath}
hence the ``total space'' of the universal [[spherical fibration]] on the [[classifying space]] $B O$ for (stable) [[real vector bundles]].

Given this, for any [[topological group]] $G$ equipped with a [[homomorphism]] to the [[orthogonal group]] there is a corresponding Thom spectrum

\begin{displaymath}
M G \simeq \underset{\longrightarrow}{\lim}(B G\to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)
  \,.
\end{displaymath}
This is considered particularly for the stages $G$ in the [[Whitehead tower]] of the [[orthogonal group]], where it yields $M$[[Spin group|Spin]], $M$[[String group]], etc.

All these Thom spectra happen to naturally have the structure of [[E-∞ rings]] and $E_\infty$-ring homomorphisms $M O\to E$ into another $E_\infty$-ring $E$ are equivalently universal [[orientation in generalized cohomology|orientations in E-cohomology]]. On [[homotopy groups]] these are [[genera]] with [[coefficients]] in the underlying ring $\pi_\bullet(E)$.

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

\hypertarget{ForVectorBundles}{}\subsubsection*{{For vector bundles}}\label{ForVectorBundles}

First recall the following two basic facts about the construction of [[Thom spaces]]. See at \emph{[[Thom space]]} \href{Thom+space#SuspensionOfThomSpaces}{this prop.}.

\begin{prop}
\label{SuspensionOfThomSpaces}\hypertarget{SuspensionOfThomSpaces}{}
For $V \to X$ a [[vector bundle]], there is a [[homeomorphism]]

\begin{displaymath}
Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V)
\end{displaymath}
between, on the one hand, the [[Thom space]] of the [[direct sum of vector bundles]] of $V$ with the trivial [[vector bundle]] of [[rank]] $n$ and, on the other, the $n$-fold [[reduced suspension]] of the [[Thom space]] of $V$.

\end{prop}
\begin{prop}
\label{ThomSpaceOfExternalProductOfVectorBundles}\hypertarget{ThomSpaceOfExternalProductOfVectorBundles}{}
For $V_1 \to X_1$ and $V_2 \to X_2$ to vector bundles, let $V_1 \boxtimes V_2 \to X_1 \times X_2$ be the [[direct sum of vector bundles]] of their [[pullbacks]] to $X_1 \times X_2$. The corresponding Thom space is the [[smash product]] of the individual Thom spaces:

\begin{displaymath}
Th(V_1 \boxtimes V_2)
  \simeq
  Th(V_1) \wedge Th(V_2)
  \,.
\end{displaymath}
\end{prop}
Prop. \ref{SuspensionOfThomSpaces} will give rise to universal Thom spectra in the following, while prop. \ref{ThomSpaceOfExternalProductOfVectorBundles} will give them the struture of [[ring spectra]].

\begin{defn}
\label{ThomSpectrumOfAVectorBundle}\hypertarget{ThomSpectrumOfAVectorBundle}{}
For $V \to X$ a [[vector bundle]], its \textbf{Thom spectrum} is the [[suspension spectrum]] $\Sigma^\infty Th(V)$ of the [[Thom space]] of $V$. By prop. prop. \ref{SuspensionOfThomSpaces} this may be written as

\begin{displaymath}
(\Sigma^\infty Th(V)) 
  \simeq
  Th(\mathbb{R}^n \oplus V)
\end{displaymath}
with structure maps are the equivalences

\begin{displaymath}
\sigma_n
    \;\colon\;
  \Sigma Th(\mathbb{R}^n \oplus V) 
    \stackrel{\simeq}{\longrightarrow} 
  Th(\mathbb{R}^{n+1} \oplus V) 
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{PullbackOfUniversalOnBundleUnderCoordinateRestriction}\hypertarget{PullbackOfUniversalOnBundleUnderCoordinateRestriction}{}
For each $n \in \mathbb{N}$ the [[pullback]] of the [[rank]]-$(n+1)$ [[universal vector bundle]] to the [[classifying space]] of rank $n$ vector bundles is the [[direct sum of vector bundles]] of the rank $n$ universal vector bundle with the trivial rank-1 bundle: there is a [[pullback]] [[diagram]] of [[topological spaces]] of the form

\begin{displaymath}
\itexarray{
    \mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)
    &\longrightarrow&
    E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}
    \\
    \downarrow &(pb)& \downarrow
    \\
    B O(n) &\longrightarrow& B O(n+1)
  }
  \,,
\end{displaymath}
where the bottom morphism is the canonical one (\href{classifying+space#InclusionOfBOnIntoBOnPlusOne}{def.}).

\end{prop}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, p. 25})

\begin{proof}
For each $k \in \mathbb{N}$, $k \geq n$ there is such a pullback of the canonical vector bundles over [[Grassmannians]]

\begin{displaymath}
\itexarray{
    \left\{
      {V_{n}\subset \mathbb{R}^k, }
      \atop
      {v \in V_n, v_{n+1} \in \mathbb{R}}
    \right\}
    &\longrightarrow&
    \left\{
      {V_{n+1} \subset \mathbb{R}^{k+1}},
      \atop
      v \in V_{n+1}
    \right\}
    \\
    \downarrow && \downarrow
    \\
    Gr_n(\mathbb{R}^k)
    &\longrightarrow&
    Gr_{n+1}(\mathbb{R}^{k+1})
  }
\end{displaymath}
where the bottom morphism is the canonical inclusion (\href{classifying+space#InclusionOfBOnIntoBOnPlusOne}{def.}). Under taking [[colimit]] over $k$, this produces the claimed pullback.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
The $n$-fold looping of the Thom spectrum, according to def. \ref{ThomSpectrumOfAVectorBundle}, of the rank-$n$ [[universal vector bundle]] is written

\begin{displaymath}
M O(n)
    \coloneqq
  \Sigma^{-n} \Sigma^\infty Th( E O(n) \underset{O(n)}{\times} \mathbb{R}^n )
  \,.
\end{displaymath}
The image of the top horizontal maps in prop. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction} under $\Sigma^{-n-1}Th(-)$ are, via prop. \ref{SuspensionOfThomSpaces}, maps of the form

\begin{displaymath}
M O(n) \longrightarrow M O(n+1)
\end{displaymath}
The [[homotopy colimit]] over these maps is the universal Thom spectrum:

\begin{displaymath}
M O \coloneqq {\lim_\to}_n M O(n) 
  \,.
\end{displaymath}
\end{defn}
More explicitly:

\begin{defn}
\label{UniversalThomSpectrum}\hypertarget{UniversalThomSpectrum}{}
As a [[sequential spectrum]], the \emph{universal Thom spectrum} $M O$ is represented by the [[sequential prespectrum]] whose $n$th component space is the [[Thom space]]

\begin{displaymath}
(M O)_n \coloneqq Th(E O(n)\underset{O(n)}{\times}\mathbb{R}^n)
\end{displaymath}
of the rank-$n$ [[universal vector bundle]], and whose structure maps are the image under the [[Thom space]] functor $Th(-)$ of the top morphisms in prop. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction}, via the homeomorphisms of prop. \ref{SuspensionOfThomSpaces}:

\begin{displaymath}
\sigma_n
   \;\colon\;
  \Sigma (M O)_n
    \simeq 
  Th(\mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n))
    \stackrel{}{\longrightarrow} 
  Th(E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}) 
    = 
  (M O)_{n+1}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
As an [[orthogonal ring spectrum]], the universal [[Thom spectrum]] $M O$ has

\begin{itemize}%
\item component spaces

\begin{displaymath}
(M O)_V \coloneqq E O(V)_+ \underset{O(V)}{\wedge} S^V
\end{displaymath}
the [[Thom spaces]] of the [[universal vector bundle]] with fiber $V$;


\item left $O(V)$-action induced by the remaining canonical left action of $E O(V)$;


\item multiplication maps

\begin{displaymath}
(E O(V_1)_+ \underset{O(V_1)}{\wedge} S^{V_1})
    \wedge
  (E O(V_2)_+ \underset{O(V_2)}{\wedge} S^{V_2}  
    \longrightarrow
  E O(V_1 \oplus V_2)_+ 
    \underset{O(V_1 \oplus V_2)}{\wedge} 
  S^{V_1 \oplus V_2}
\end{displaymath}
induced via prop. \ref{ThomSpaceOfExternalProductOfVectorBundles}


\item unit maps given by

\begin{displaymath}
S^V \simeq O(V)_+ \wedge_{O(V)} S^V
    \longrightarrow
  E O(V)_+ \wedge_{O(V)} S^V
\end{displaymath}
induced by the fiber inclusion $O(V) \to E O(V)$.



\end{itemize}
\end{defn}
For discussion of the refinement of the Thom spectrum $M O$ to a [[symmetric spectrum]] see (\hyperlink{HoveyShipleySmith00}{Hovey-Shipley-Smith 00, example 6.2.3}, \hyperlink{Schwede12}{Schwede 12, Example I.2.8}). For the refinement to an [[orthogonal spectrum]] and [[globally equivariant spectrum]] see (\hyperlink{Schwede15}{Schwede 15, section V.4}).

More generally, there are universal Thom spectra associated with any other tangent structure (``[[(B,f)]-structure]''), notably for the orthogonal group replaced by the [[special orthogonal groups]] $SO(n)$, or the [[spin groups]] $Spin(n)$, or the [[string 2-group]] $String(n)$, or the [[fivebrane 6-group]] $Fivebrane(n)$,\ldots{}, or any level in the [[Whitehead tower]] of $O(n)$. To any of these groups there corresponds a Thom spectrum (denoted, respectively, $M SO$, $M Spin$, $M String$, $M Fivebrane$, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.

Recall:

\begin{defn}
\label{BfStructure}\hypertarget{BfStructure}{}
A [[(B,f)-structure]] $\mathcal{B}$, is a system of [[Serre fibrations]]

\begin{displaymath}
\itexarray{
    B_n
    \\
    \downarrow^{\mathrlap{f_n}}
    \\
    B O(n)
  }
\end{displaymath}
for $n \in \mathbb{N}$, equipped with maps $j_n \colon B_n \to B_{n+1}$ covering the canonical maps $i_n \colon B O(n) \to B O(n+1)$ ((\href{classifying+space#InclusionOfBOnIntoBOnPlusOne}{def.})) in that there are [[commuting squares]]

\begin{displaymath}
\itexarray{
    B_n &\overset{j_n}{\longrightarrow}& B_{n+1}
    \\
    \downarrow^{\mathrlap{f_n}}
      &&
    \downarrow^{\mathrlap{f_{n+1}}}
    \\
    B O(n)
      &\overset{i_n}{\longrightarrow}&
    B O(n+1)
  }
  \,.
\end{displaymath}
Similarly, an \textbf{$S^2$-$(B,f)$-structure} is a compatible system

\begin{displaymath}
f_{2n} \colon B_{2n} \longrightarrow B O(2n)
\end{displaymath}
indexed only on the even natural numbers.

\end{defn}
\begin{defn}
\label{VectorBundleAssociatedWithBfStructure}\hypertarget{VectorBundleAssociatedWithBfStructure}{}
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. \ref{BfStructure}), write

\begin{displaymath}
E^{\mathcal{B}}_n \coloneqq f_n^\ast ( E O(n) \underset{O(n)}{\times} \mathbb{R}^n )
\end{displaymath}
for the [[pullback]] of the rank-$n$ [[universal vector bundle]] from $B O(n)$ to $B_n$ along $f_n$.

\end{defn}
Observe that the analog of prop. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction} still holds:

\begin{prop}
\label{PullbackOfUniversalBfBundleUnderCoordinateRestriction}\hypertarget{PullbackOfUniversalBfBundleUnderCoordinateRestriction}{}
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. \ref{BfStructure}), then the pullback of its rank-$(n+1)$ vector bundle $E^{\mathcal{B}}_{n+1}$ (def. \ref{VectorBundleAssociatedWithBfStructure}) along the map $j_n \colon B_n \to B_{n+1}$ is the [[direct sum of vector bundles]] of the rank-$n$ bundle $E^{\mathcal{B}}_n$ with the trivial rank-1-bundle: there is a pullback square

\begin{displaymath}
\itexarray{
    \mathbb{R} \oplus E^{\mathcal{B}}_n 
      &\longrightarrow&
    E^{\mathcal{B}}_{n+1}
    \\
    \downarrow &(pb)& \downarrow
    \\
    B_n &\underset{j_n}{\longrightarrow}& B_{n+1}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Unwinding the definitions, the pullback in question is

\begin{displaymath}
\begin{aligned}
    j_n^\ast E^{\mathcal{B}}_{n+1}
      & =
    j_n^\ast f_{n+1}^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1})
    \\
    & \simeq
    (j_n \circ f_{n+1})^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1})
    \\
    & \simeq
    ( f_n \circ i_n )^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1})
    \\
    & \simeq
    f_n^\ast i_n^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1})
    \\
    & \simeq
    f_n^\ast (\mathbb{R} \oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^{n}))
    \\
    &\simeq \mathbb{R} \oplus E^{\mathcal{B}_n}
    \,,
  \end{aligned}
\end{displaymath}
where the second but last step is due to prop. \ref{PullbackOfUniversalOnBundleUnderCoordinateRestriction}.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
Given a [[(B,f)-structure]] $\mathcal{B}$ (def. \ref{BfStructure}), its Thom spectrum $M \mathcal{B}$ is, as a [[sequential prespectrum]], given by component spaces being the [[Thom spaces]] of the $\mathcal{B}$-associated vector bundles of def. \ref{VectorBundleAssociatedWithBfStructure}

\begin{displaymath}
(M \mathcal{B})_n
  \coloneqq
  Th(E^{\mathcal{B}}_n)
\end{displaymath}
and with structure maps given via prop. \ref{SuspensionOfThomSpaces} by the top maps in prop. \ref{PullbackOfUniversalBfBundleUnderCoordinateRestriction}:

\begin{displaymath}
\sigma_n
    \;\colon\;
  \Sigma (M \mathcal{B})_n
    =
  \Sigma Th(E^{\mathcal{E}}_n)
    \simeq
  Th(\mathbb{R}\oplus E^{\mathcal{E}}_n)
    \longrightarrow
  Th(E^{\mathcal{E}}_{n+1})
  =
  (M \mathcal{B})_{n+1}
  \,.
\end{displaymath}
Similarly for a $(B,f)$-structure indexed on the even natural numbers, there is the corresponding Thom spectrum as an $S^2$-sequential spectrum (\href{model+structure+on+topological+sequential+spectra#SequentialS2Spectra}{def.}).

If $B_n = B G_n$ for some natural system of groups $G_n \to O(n)$, then one usually writes $M G$ for $M \mathcal{B}$. For instance $M SO$, $M Spin$, $M U$ etc.

If the $(B,f)$-structure is multiplicative (def. \ref{BfStructure}), then the Thom spectrum $M \mathcal{B}$ canonical becomes a [[ring spectrum]]: the multiplication maps $B_{n_1} \times B_{n_2}\to B_{n_1 + n_2}$ are covered by maps of vector bundles

\begin{displaymath}
E^{\mathcal{B}}_{n_1} \boxtimes E^{\mathcal{B}}_{n_2}
  \longrightarrow
  E^{\mathcal{B}}_{n_1 + n_2}
\end{displaymath}
and under forming [[Thom spaces]] this yields, by prop. \ref{ThomSpaceOfExternalProductOfVectorBundles}, maps

\begin{displaymath}
(M \mathcal{B})_{n_1}
  \wedge
  (M \mathcal{B})_{n_2}
  \longrightarrow
  (M \mathcal{B})_{n_1 + n_2}
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The Thom spectrum of the \emph{[[framing]] structure} (\href{G-structure#ExamplesOfBfStructures}{exmpl.}) is equivalently the [[sphere spectrum]]

\begin{displaymath}
M 1 \simeq \mathbb{S}
  \,.
\end{displaymath}
Because in this case $B_n \simeq \ast$ and so $E^{\mathcal{B}}_n \simeq \mathbb{R}^n$, whence $Th(E^{\mathcal{B}}_n) \simeq S^n$.

\end{example}
\hypertarget{ForSphereBundles}{}\subsubsection*{{For spherical fibrations}}\label{ForSphereBundles}

\begin{defn}
\label{ThomSpectrumForSphericalFibrations}\hypertarget{ThomSpectrumForSphericalFibrations}{}
(\ldots{})

\end{defn}
\hypertarget{ForInfinityModuleBundles}{}\subsubsection*{{For $(\infty,1)$-module bundles}}\label{ForInfinityModuleBundles}

We discuss the Thom spectrum construction for general [[(∞,1)-module bundles]].

\begin{prop}
\label{}\hypertarget{}{}
There is pair of [[adjoint (∞,1)-functors]]

\begin{displaymath}
(\Sigma^\infty \Omega^\infty \dashv gl_1)
   \colon
  E_\infty Rings
   \stackrel{\overset{\Sigma^\infty \Omega^\infty}{\leftarrow}}{\underset{gl_1}{\to}}
  Spec_{con}
  \,,
\end{displaymath}
where $(\Sigma^\infty \dashv \Omega^\infty) : Spec \to Top$ is the [[stabilization]] adjunction between [[Top]] and [[Spec]] ($\Sigma^\infty$ forms the [[suspension spectrum]]), restricted to connective spectra. The [[right adjoint]] is the \emph{[[∞-group of units]]}-[[(∞,1)-functor]], see there for more details.

\end{prop}
This is (\hyperlink{ABGHR}{ABGHR, theorem 2.1/3.2}).

\begin{remark}
\label{}\hypertarget{}{}
Here $gl_1$ forms the ``[[general linear group]]-of rank 1''-spectrum of an [[E-∞ ring]]: its [[∞-group of units]]``. The adjunction is the generalization of the adjunction

\begin{displaymath}
(\mathbb{Z}[-] \dashv GL_1) : 
   CRing \stackrel{\overset{\mathbb{Z}[1]}{\leftarrow}}{\underset{GL_1}{\to}}
  Ab
\end{displaymath}
between [[CRing]] and [[Ab]], where $\mathbb{Z}[-]$ forms the [[group ring]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
b gl_1(R) \coloneqq \Sigma gl_1(R)
\end{displaymath}
for the [[suspension]] of the group of units $gl_1(R)$.

\end{defn}
This plays the role of the [[classifying space]] for $gl_1(R)$-[[principal ∞-bundle]]s.

For $f : b \to b gl_1(R)$ a morphism (a [[cocycle]] for $gl_1(R)$-bundles) in [[Spec]], write $p \to b$ for the corresponding bundle: the [[homotopy fiber]]

\begin{displaymath}
\itexarray{
     p &\to& * 
     \\
     \downarrow && \downarrow
     \\
     b &\stackrel{f}{\to}& b gl_1(R)
  }
  \,.
\end{displaymath}
Given a $R$-algebra $A$, hence an [[A-∞ algebra]] over $R$, exhibited by a morphism $\rho : R \to A$, the composite

\begin{displaymath}
\rho(f) : b \stackrel{f}{\to} b gl_1(R) \stackrel{\rho}{\to} b gl_1(A)
\end{displaymath}
is that for the corresponding [[associated ∞-bundle]].

We write capital letters for the underlying spaces of these spectra:

\begin{displaymath}
P \coloneqq \Sigma^\infty \Omega^\infty p
\end{displaymath}
\begin{displaymath}
B \coloneqq \Sigma^\infty \Omega^\infty b
\end{displaymath}
\begin{displaymath}
GL_1(R) \coloneqq \Sigma^\infty \Omega^\infty gl_1(R)
\end{displaymath}
\begin{defn}
\label{GeneralThomSpectrum}\hypertarget{GeneralThomSpectrum}{}
The \textbf{Thom spectrum} $M f$ of $f : b \to gl_1(R)$ is the [[(∞,1)-pushout]]

\begin{displaymath}
\itexarray{
    \Sigma^\infty \Omega^\infty R &\to& R
    \\
    \downarrow && \downarrow
    \\
    \Sigma^\infty \Omega^\infty p &\to& M f
  }
  \,,
\end{displaymath}
hence the [[derived functor|derived]] [[smash product]]

\begin{displaymath}
M f \simeq P \wedge_{GL_1(R)} R
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This means that a morphism $M f \to A$ is an $GL_1(R)$-equivariant map $P \to A$.

Notice that for $R = \mathbb{C}$ the [[complex number]]s, $B \to GL_1(R)$ is the cocycle for a [[circle bundle]] $P \to B$. A $U(1)$-equivariant morphism $P \to A$ to some [[representation]] $A$ is equivalently a [[section]] of the A-[[associated bundle]].

Therefore the Thom spectrum may be thought of as co-[[representable functor|representing]] spaces of sections of associated bundles

``$Hom(M f, A) \simeq \Gamma(P \wedge_{GL_1(R)} A)$''.

\end{remark}
This is made precise by the following statement.

\begin{prop}
\label{}\hypertarget{}{}
We have an [[(∞,1)-pullback]] diagram

\begin{displaymath}
\itexarray{
    E_\infty Alg_R(M f, A) &\stackrel{}{\to}& (...)
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{}{\to}& (...)
  }
\end{displaymath}
\end{prop}
This is (\hyperlink{ABGHR}{ABGHR, theorem 2.10}).

This definition does subsume the \href{ForSphereBundles}{above} definition of Thom spectra for [[sphere bundles]] (hence also that for [[vector bundles]], by removing their [[zero section]]):

\begin{prop}
\label{}\hypertarget{}{}
Let $R = S$ be the [[sphere spectrum]]. Then for $f \colon b \to gl_1(S)$ a cocycle for an $S$-bundle,

\begin{displaymath}
G \coloneqq \Omega^\infty g \colon B \to B GL_1(S)
\end{displaymath}
is the classifying map for a [[spherical fibration]] over $B \in Top$.

The Thom spectrum $M f$ of def. \ref{GeneralThomSpectrum} is equivalent to the Thom spectrum of the spherical fibration, according to def. \ref{ThomSpectrumForSphericalFibrations}.

\end{prop}
This is in (\hyperlink{ABGHR}{ABGHR, section 8}).

Equivalently the Thom spectrum is characterized as follows:

\begin{defn}
\label{ThomSpectrumAsHomotopyColimit}\hypertarget{ThomSpectrumAsHomotopyColimit}{}
For $\chi \colon X \to  R Line$ a map to the [[∞-group]] of $R$-[[(∞,1)-lines]] inside $R Mod$, the corresponding Thom spectrum is the [[(∞,1)-colimit]]

\begin{displaymath}
\Gamma(\chi) \coloneqq
  \underset{\rightarrow}{\lim}
  \left(
    X \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod
  \right)
  \,.
\end{displaymath}
This construction evidently extendes to an [[(∞,1)-functor]]

\begin{displaymath}
\Gamma \colon \infty Grpd_{/R Line} \to R Mod
  \,.
\end{displaymath}
\end{defn}
This is (\hyperlink{ABG10}{Ando-Blumberg-Gepner 10, def. 4.1}), reviewed also as (\hyperlink{Wilson13}{Wilson 13, def. 3.3}).

\begin{remark}
\label{}\hypertarget{}{}
This is the $R$-[[(∞,1)-module]] of [[sections]] of the [[(∞,1)-module bundle]] classified by $X \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod$.

By the [[universal property]] of the [[(∞,1)-colimit]] we have for $\underline{R} \colon X \to R Mod$ the trivial $R$-bundle that

\begin{displaymath}
Hom_{[X, R Mod]}(\chi, \underline{R})
  \simeq
  Hom_{R Mod}(\Gamma(\chi), R)
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The section/Thom spectrum functor is the left [[(∞,1)-Kan extension]] of the canonical embedding $R Line \hookrightarrow R Mod$ along the [[(∞,1)-Yoneda embedding]]

\begin{displaymath}
R Line \hookrightarrow [R Line^{op}, \infty Grpd]
  \simeq \infty Grpd_{/R Line}
\end{displaymath}
(where the [[equivalence of (∞,1)-categories]] on the right is given by the [[(∞,1)-Grothendieck construction]]). In other words, it is the essentially unique [[(∞,1)-colimit]]-preserving [[(∞,1)-functor]] $\infty Grpd_{/ R Line} \to R Mod$ which restricts along this inclusion to the canonical embedding.

\end{prop}
This observation appears as (\hyperlink{Wilson13}{Wilson 13, prop. 4.4}).

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

\hypertarget{RingSpectrumStructure}{}\subsubsection*{{Ring spectrum structure}}\label{RingSpectrumStructure}

The universal Thom spectrum, def. \ref{UniversalThomSpectrum}, naturally inherits the structure of a [[ring spectrum]] as follows.

\begin{prop}
\label{}\hypertarget{}{}
There are canonical commuting diagrams

\begin{displaymath}
\itexarray{
    ( E (O(k)\times O(\ell))\underset{O(k)\times O(\ell)}{\times}) 
    \mathbb{R}^k \oplus \mathbb{R}^\ell
    &\longrightarrow& 
    E O(k+\ell)\underset{O(k + ell)}{\times} \mathbb{R}^{k+\ell}
    \\
    \downarrow && \downarrow
    \\
    B O(k)\times B O(\ell)
    &\stackrel{}{\longrightarrow}&
    B O(k + \ell)
  }
  \,.
\end{displaymath}
Applying the [[Thom space]] functor to the top morphisms and using prop. \ref{ThomSpaceOfExternalProductOfVectorBundles} gives morphisms

\begin{displaymath}
M O(k) \wedge M O(\ell)
  \longrightarrow
  M O(k + \ell)
\end{displaymath}
that combine to a [[functor with smash products]] and hence give $M O$ the structure of a [[ring spectrum]].

\end{prop}
More abstractly, sufficient condition for a Thom spectrum of an ∞-module bundle (as \hyperlink{ForInfinityModuleBundles}{above}) to have [[E-∞ ring]] structure is that it arises, as the [[(∞,1)-colimit]] of a homomorphism of [[E-∞ spaces]] $B \to A Line$ (\hyperlink{ABG10}{ABG 10, prop.6.21}).

\hypertarget{relation_to_the_cobordism_ring}{}\subsubsection*{{Relation to the cobordism ring}}\label{relation_to_the_cobordism_ring}

\begin{prop}
\label{}\hypertarget{}{}
The [[cobordism ring|cobordism group]] of un[[oriented]] $n$-[[dimension]]al [[manifold]]s is [[natural isomorphism|naturally isomorphic]] to the $n$th [[homotopy group]] of the Thom spectrum $M O$. That is, there is a natural isomorphism

\begin{displaymath}
\Omega^{un}_\bullet \simeq \pi_\bullet M O 
   \coloneqq
  {\lim_{\to}}_{k \to \infty} \pi_{n+k} M O(k)
  \,.
\end{displaymath}
\end{prop}
This is a seminal result due to (\hyperlink{Thom54}{Thom 54}), whose proof proceeds by the [[Pontryagin-Thom construction]]. The presentation of the following proof follows (\hyperlink{Francis3}{Francis, lecture 3}).

\begin{proof}
We first construct a map $\Theta : \Omega_n^{un} \to \pi_n M O$.

Given a class $[X] \in \Omega_n^{un}$ we can choose a representative $X \in$ [[SmthMfd]] and a [[closed embedding]] $\nu$ of $X$ into the [[Cartesian space]] $\mathbb{R}^{n+k}$ of sufficiently large [[dimension]]. By the [[tubular neighbourhood theorem]] $\nu$ factors as the embedding of the [[zero section]] into the [[normal bundle]] $N_\nu$ followed by an [[open embedding]] of $N_\nu$ into $\mathbb{R}^{n+k}$

\begin{displaymath}
\itexarray{
     X &&\stackrel{\nu}{\hookrightarrow}&& \mathbb{R}^{n+k}
     \\
     & \searrow && \nearrow_{\mathrlap{i}}
     \\
     && N_\nu
  }
  \,.
\end{displaymath}
Now use the [[Pontrjagin-Thom construction]] to produce an element of the [[homotopy group]] first in the [[Thom space]] $Th(N_\nu)$ of $N_\nu$ and then eventually in $M O$. To that end, let

\begin{displaymath}
\mathbb{R}^{n+k} \to (\mathbb{R}^{n+k})^+ \simeq S^{n+k}
\end{displaymath}
be the map into the [[one-point compactification]]. Define a map

\begin{displaymath}
t : S^{n+k} \simeq (\mathbb{R}^{n+k})^+ \to Disk(N_\nu)/Sphere(N_\nu) \simeq Th(N_\nu)
\end{displaymath}
by sending points in the image of $Disk(N_\nu)$ under $i$ to their preimage, and all other points to the collapsed point $Sphere(N_\nu)$. This defines an element in the [[homotopy group]] $\pi_{n+k}(Th(N_\nu))$.

To turn this into an element in the homotopy group of $M O$, notice that since $N_\nu$ is a [[vector bundle]] of [[rank]] $k$, it is the [[pullback]] by a map $\mu$ of the universal rank $k$ vector bundle $\gamma_k \to B O(k)$

\begin{displaymath}
\itexarray{
    N_\nu \simeq \mu^* \gamma_k &\to& \gamma_k
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\mu}{\to}& B O(k)
  }
  \,.
\end{displaymath}
By forming Thom spaces the top map induces a map

\begin{displaymath}
Th(N_\nu) \to Th(\gamma^k) =: M O(k)
  \,.
\end{displaymath}
Its composite with the map $t$ constructed above gives an element in $\pi_{n+k} M O(k)$

\begin{displaymath}
S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)
\end{displaymath}
and by $\pi_{n+k} M O(k) \to {\lim_\to}_k \pi_{n+k} M O(k) =: \pi_n M O$ this is finally an element

\begin{displaymath}
\Theta
  :
  [X]
   \mapsto
  (S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)) 
  \in
  \pi_n M O
  \,.
\end{displaymath}
We show now that this element does not depend on the choice of embedding $\nu : X \to \mathbb{R}^{n+k}$.

(\ldots{})

Finally, to show that $\Theta$ is an [[isomorphism]] by constructing an inverse.

For that, observe that the [[sphere]] $S^{n+k}$ is a [[compact topological space]] and in fact a [[compact object]] in [[Top]]. This implies that every map $f$ from $S^{n+k}$ into the [[filtered colimit]]

\begin{displaymath}
Th(\gamma^k)  \simeq {\lim_\to}_s Th(\gamma^k_s)
  \,,
\end{displaymath}
factors through one of the terms as

\begin{displaymath}
f : S^{n+k} \to Th(\gamma^k_s) \hookrightarrow Th(\gamma^k)
  \,.
\end{displaymath}
By [[Thom's transversality theorem]] we may find an embedding $j : Gr_k(\mathbb{R}^s) \to Th(\gamma^k_s)$ by a [[transverse map]] to $f$. Define then $X$ to be the [[pullback]]

\begin{displaymath}
\itexarray{
    X &\to& Gr_k(\mathbb{R}^s)
    \\
    \downarrow && \downarrow^{\mathrlap{j}}
    \\
    S^{n+k} &\stackrel{f}{\to}& Th(\gamma^k_s)
  }
  \,.
\end{displaymath}
We check that this construction provides an inverse to $\Theta$.

(\ldots{})

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The [[homotopy equivalence]] $\Omega^\infty M O \simeq \vert Cob_\infty \vert$ is the content of the [[Galatius-Madsen-Tillmann-Weiss theorem]], and is now seen as a part of the [[cobordism hypothesis]] theorem.

\end{remark}
\hypertarget{AsDualObject}{}\subsubsection*{{As a dual in the stable homotopy category}}\label{AsDualObject}

Write [[Spec]] for the category of [[spectra]] and $Ho(Spec)$ for its standard [[homotopy category]]: the [[stable homotopy category]]. By the [[symmetric monoidal smash product of spectra]] this becomes a [[monoidal category]].

For $X$ any [[topological space]], we may regard it as an object in $Ho(Spec)$ by forming its [[suspension spectrum]] $\Sigma^\infty_+ X$. We may ask under which conditions on $X$ this is a [[dualizable object]] with respect to the smash-product monoidal structure.

It turns out that a sufficient condition is that $X$ a closed [[smooth manifold]] or more generally a [[compact topological space|compact]] Euclidean [[neighbourhood retract]]. In that case $Th(N X)$ -- the Thom spectrum of its [[normal bundle|stable normal bundle]] is the corresponding [[dual object]]. (\hyperlink{Atiyah61}{Atiyah 61}, \hyperlink{DoldPuppe78}{Dold-Puppe 78}). This is called the [[Spanier-Whitehead dual]] of $\Sigma^\infty_+ X$.

Using this one shows that the [[trace]] of the identity on $\Sigma^\infty_+ X$ in $Ho(Spec)$ -- the categorical [[dimension]] of $\Sigma^\infty_+ X$ -- is the [[Euler characteristic]] of $X$.

For a brief exposition see (\hyperlink{PontoShulman}{PontoShulman, example 3.7}). For more see at \emph{[[Spanier-Whitehead duality]]}.

\hypertarget{AsTheUniversalSphericalFibration}{}\subsubsection*{{As the universal spherical fibration, from the $J$-homomorphism}}\label{AsTheUniversalSphericalFibration}

The [[J-homomorphism]] is a canonical map $B O \to B gl_1(\mathbb{S})$ from the [[classifying space]] of the [[stable orthogonal group]] to the [[delooping]] of the [[infinity-group of units]] of the [[sphere spectrum]]. This classifies an ``[[(∞,1)-vector bundle]]'' of [[sphere spectrum]]-[[module spectrum|modules]] over $B O$ and this is the Thom spectrum.

So in terms of the [[(∞,1)-colimit]] description \hyperlink{ForInfinityModuleBundles}{above} we have

\begin{displaymath}
M O \simeq  \underset{\longrightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to \mathbb{S}Mod = Spectra)
  \,.
\end{displaymath}
See at \emph{[[orientation in generalized cohomology]]} for more on this.

\hypertarget{AsTheInfiniteCobordismCategory}{}\subsubsection*{{As the infinite cobordism category}}\label{AsTheInfiniteCobordismCategory}

The [[geometric realization]] for the [[(infinity,n)-category of cobordisms]] for $n \to \infty$ is the Thom spectrum

\begin{displaymath}
\vert Bord_\infty \vert \simeq \Omega^\infty MO
  \,.
\end{displaymath}
This is implied by the [[Galatius-Madsen-Tillmann-Weiss theorem]] and by [[Jacob Lurie]]`s proof of the [[cobordism hypothesis]]. See also (\href{Francis3}{Francis-Gwilliam, remark 0.9}).

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

Under the [[Brown representability theorem]] the Thom spectrum represents the [[generalized (Eilenberg-Steenrod) cohomology]] theory called [[complex cobordism cohomology theory|cobordism cohomology theory]].

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

\begin{itemize}%
\item [[cobordism theory]]


\item [[Thom space]], [[Thom isomorphism]], [[Pontryagin-Thom collapse map]]


\item [[cobordism ring]]



\end{itemize}
[[!include generalized fiber integration synonyms - table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The relation between the homotopy groups of the Thom spectrum and the cobordism ring is due to

\begin{itemize}%
\item [[René Thom]], \emph{Quelques propri\'e{}t\'e{}s globales des vari\'e{}t\'e{}s diff\'e{}rentiables} Comment. Math. Helv. 28, (1954). 17-86

\end{itemize}
Further original articles include

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Thom complexes}, Proc. London Math. Soc. (3), 11:291--310, 1961. 10 (\href{http://www.maths.ed.ac.uk/~aar/papers/atiyahthom.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Frank Adams]], part III, section 2 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Stanley Kochman]], section 1.5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[John Francis]], \emph{Topology of manifolds} course notes (2010) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/}{web})

Lecture 3 \emph{Thom's theorem} (notes by A. Smith) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf}{pdf})

A remark of the relation of the Thom spectrum to [[(∞,n)-category of cobordisms]] for $n = \infty$ is in:

Lecture 2 \emph{Cobordisms} (notes by [[Owen Gwilliam]]) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf}{pdf})


\item [[Johannes Ebert]], \emph{A lecture course on Cobordism Theory}, 2012 (\href{http://wwwmath.uni-muenster.de/u/jeber_02/skripten/bordism-skript.pdf}{pdf})


\item [[Dan Freed]], lecture 10 of \emph{Bordism: old and new}, 2013 (\href{http://www.ma.utexas.edu/users/dafr/bordism.pdf}{pdf})



\end{itemize}
As [[symmetric spectra]]:

\begin{itemize}%
\item [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], example 6.2.3 of \emph{Symmetric spectra}, J. Amer. Math. Soc. 13 (2000), 149-208 (\href{http://arxiv.org/abs/math/9801077}{arXiv:math/9801077})


\item [[Christian Schlichtkrull]], \emph{Thom Spectra that Are Symmetric Spectra}, Documenta Mathematica 14 (2009) 699-748 (\href{http://arxiv.org/abs/0811.0592}{arXiv:0811.0592})


\item [[Stefan Schwede]], Example I.2.6 in \emph{Symmetric spectra}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec.pdf}{pdf})



\end{itemize}
As [[orthogonal spectra]] and as [[equivariant spectra]]

\begin{itemize}%
\item [[Stefan Schwede]], chapter V.4 of \emph{[[Global homotopy theory]]}, 2015

\end{itemize}
Textbook discussion with an eye towards the [[generalized (Eilenberg-Steenrod) cohomology]] of [[topological K-theory]] and [[cobordism cohomology theory]] is in

\begin{itemize}%
\item [[Yuli Rudyak]], \emph{On Thom spectra, orientability and cobordism}, Springer Monographs in Mathematics, 1998 (\href{http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Frank Quinn]], \emph{Assembly maps in bordism-type theories} (\href{http://www.maths.ed.ac.uk/~aar/papers/quinnass.pdf}{pdf})

\end{itemize}
A generalized notion of Thom spectra in terms of [[(∞,1)-module bundles]] is discussed 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})

\end{itemize}
a streamlined update of which is

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], \emph{An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology} (\href{http://arxiv.org/abs/1403.4325}{arXiv:1403.4325})

\end{itemize}
Discussion of Thom spectra from the point of view of [[(∞,1)-module bundles]] is in

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \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}
which is reviewed in

\begin{itemize}%
\item [[Dylan Wilson]], \emph{Thom spectra from the $\infty$ point of view}, 2013 (\href{http://www.math.northwestern.edu/~bwill/thom/DWthom4.pdf}{pdf})

\end{itemize}
and in the context of [[motivic quantization]] via [[fiber integration in generalized cohomology|pushforward]] in [[twisted cohomology|twisted]] [[generalized cohomology]] in section 3.1 of

\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]}, master thesis 2013

\end{itemize}
\hypertarget{as_dual_objects_in_the_stable_homotopy_category}{}\subsubsection*{{As dual objects in the stable homotopy category}}\label{as_dual_objects_in_the_stable_homotopy_category}

The relation of Thom spectra to [[dualizable objects]] in the [[stable homotopy category]] is originally due to (\hyperlink{Atiyah61}{Atiyah 61}) and

\begin{itemize}%
\item [[Albrecht Dold]], [[Dieter Puppe]], \emph{Duality, trace, and transfer}. In Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pages 81--102, Warsaw, 1980. PWN.


\item [[L. Gaunce Lewis, Jr.]], [[Peter May]], M. Steinberger, and J. E. McClure, \emph{Equivariant stable homotopy theory}, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.



\end{itemize}
A brief exposition appears as example 3.7 in

\begin{itemize}%
\item [[Kate Ponto]] and [[Mike Shulman]], \emph{Traces in symmetric monoidal categories} (\href{http://arxiv.org/abs/1107.6032}{arXiv:1107.6032}, \href{http://www.sandiego.edu/~shulman/papers/ccrtraces.pdf}{slides}).

\end{itemize}
[[!redirects Thom spectra]]

[[!redirects generalized Thom spectrum]] [[!redirects generalized Thom spectra]]



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