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\begin{document}

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\section*{Thom space}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - 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{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{HomotopyTheoreticNature}{Homotopy-theoretic nature}\dotfill \pageref*{HomotopyTheoreticNature} \linebreak
\noindent\hyperlink{behaviour_under_direct_sum_of_vector_bundles}{Behaviour under direct sum of vector bundles}\dotfill \pageref*{behaviour_under_direct_sum_of_vector_bundles} \linebreak
\noindent\hyperlink{cwstructure}{CW-structure}\dotfill \pageref*{cwstructure} \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


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

The \textbf{Thom space} $Th(V)$ of a real [[vector bundle]] $V \to X$ over a [[topological space]] $X$ is the [[topological space]] obtained by first forming the disk bundle $D(V)$ of (unit) disks in the [[fibers]] of $V$ (with respect to a [[metric]] given by any choice of [[orthogonal structure]]) and then identifying to a point the [[boundaries]] of all the disks, i.e. forming the [[quotient topological space]] by the [[unit sphere bundle]] $S(V)$:

\begin{displaymath}
Th(V) \coloneqq D(V)/S(V)
  \,.
\end{displaymath}
(N.B.: this is a quotient of the \emph{total} spaces of the bundles taken in $Top$, not a bundle quotient in $Top/V$.)

This is equivalently the [[mapping cone]]

\begin{displaymath}
\itexarray{
    S(V) &\longrightarrow& *
    \\
    {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow
    \\
    X & \longrightarrow & Th(V)
  }
\end{displaymath}
in [[Top]] of the [[sphere]] [[bundle]] of $V$. Therefore more generally, for $P \to X$ any [[n-sphere]]-[[fiber bundle]] over $X$ ([[spherical fibration]]), its Thom space is the the [[mapping cone]]

\begin{displaymath}
\itexarray{
    P &\longrightarrow& *
    \\
    {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow
    \\
    X & \longrightarrow & Th(P)
  }
\end{displaymath}
of the bundle projection.

For $X$ a [[compact topological space]], $Th(V)$ is a model for [[generalized the|the]] [[one-point compactification]] of the total space $V$.

The Thom space of the rank-$n$ [[universal vector bundle]] over the [[classifying space]] $B O(n)$ of the [[orthogonal group]] is usually denoted $M O(n)$. As $n$ ranges, these spaces form the [[Thom spectrum]].

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

\begin{defn}
\label{ThomSpace}\hypertarget{ThomSpace}{}
Let $X$ be a [[topological space]] and let $V \to X$ be a [[vector bundle]] ([[topological vector bundle]]) over $X$ of [[rank]] $n$, which is [[associated bundle|associated]] to an [[orthogonal group|O(n)]]-[[principal bundle]]. Equivalently this means that $V \to X$ is the [[pullback]] of the [[universal vector bundle]] $E_n \to B O(n)$ over the [[classifying space]]. Since $O(n)$ preserves the [[metric]] on $\mathbb{R}^n$, by definition, such $V$ inherits the structure of a [[metric space]]-[[fiber bundle]]. With respect to this structure:c

\begin{enumerate}%
\item the \textbf{unit disk bundle} $D(V) \to X$ is the subbundle of elements of [[norm]] $\leq 1$;


\item the \textbf{[[unit sphere]] bundle} $S(V)\to X$ is the subbundle of elements of norm $= 1$;

$S(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V$;


\item the \textbf{Thom space} $Th(V)$ is the [[cofiber]] (formed in [[Top]] (\href{Top#DescriptionOfLimitsAndColimitsInTop}{prop.})) of $i_V$

\begin{displaymath}
Th(V) \coloneqq cofib(i_V)
\end{displaymath}
canonically regarded as a [[pointed topological space]].



\end{enumerate}
\begin{displaymath}
\itexarray{
    S(V) &\overset{i_V}{\longrightarrow}& D(V)
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& Th(V)
  }
  \,.
\end{displaymath}
If $V \to X$ is a general real vector bundle, then there exists an isomorphism to an $O(n)$-[[associated bundle]] and the Thom space of $V$ is, up to based [[homeomorphism]], that of this orthogonal bundle.

\end{defn}
\begin{remark}
\label{ThomSpaceForRankZeroBundle}\hypertarget{ThomSpaceForRankZeroBundle}{}
If the [[rank]] of $V$ is positive, then $S(V)$ is non-empty and then the Thom space is the [[quotient topological space]]

\begin{displaymath}
Th(V) \simeq D(V)/S(V)
  \,.
\end{displaymath}
However, in the degenerate case that the [[rank]] of $V$ vanishes, hence the case that $V = X\times \mathbb{R}^0 \simeq X$, then $D(V) \simeq V \simeq X$, but $S(V) = \emptyset$. Hence now the [[pushout]] defining the cofiber is

\begin{displaymath}
\itexarray{
    \emptyset &\overset{i_V}{\longrightarrow}& X
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& Th(V) \simeq X_*
  }
  \,,
\end{displaymath}
which exhibits $Th(V)$ as the [[coproduct]] of $X$ with the point, hence as $X$ with a basepoint freely adjoined.

\begin{displaymath}
Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+
  \,.
\end{displaymath}
\end{remark}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{HomotopyTheoreticNature}{}\subsubsection*{{Homotopy-theoretic nature}}\label{HomotopyTheoreticNature}

\begin{prop}
\label{ThomSpaceOverCWComplexIsHomotopyCofiber}\hypertarget{ThomSpaceOverCWComplexIsHomotopyCofiber}{}
Let $V \to X$ be a [[vector bundle]] over a [[CW-complex]] $X$. Then the Thom space $Th(V)$ (def. \ref{ThomSpace}) is equivalently the [[homotopy cofiber]] (\href{Introduction+to+Stable+homotopy+theory+--+P#HomotopyFiber}{def.}) of the inclusion $S(V) \longrightarrow D(V)$ of the sphere bundle into the disk bundle.

\end{prop}
\begin{proof}
The Thom space is defined as the ordinary [[cofiber]] of $S(V)\to D(V)$. Under the given assumption, this inclusion is a [[relative cell complex]] inclusion, hence a cofibration in the [[classical model structure on topological spaces]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopQuillenModelStructure}{thm.}). Therefore in this case the ordinary cofiber represents the homotopy cofiber (\hyperlink{Introduction+to+Stable+homotopy+theory+--+P#HomotopyFiber}{def.}).

\end{proof}
The equivalence to the following alternative model for this homotopy cofiber is relevant when discussing [[Thom isomorphisms]] and [[orientation in generalized cohomology]]:

\begin{prop}
\label{ThomSpaceOverCWEquivalentToConeOnInclusionOfComplementOf0Section}\hypertarget{ThomSpaceOverCWEquivalentToConeOnInclusionOfComplementOf0Section}{}
Let $V \to X$ be a [[vector bundle]] over a [[CW-complex]] $X$. Write $V \setminus X$ for the [[complement]] of its 0-[[section]]. Then the Thom space $Th(V)$ (def. \ref{ThomSpace}) is [[homotopy equivalence|homotopy equivalent]] to the [[mapping cone]] of the inclusion $(V \setminus X) \hookrightarrow V$ (hence to the pair $(V,V \setminus X)$ in the language of [[generalized (Eilenberg-Steenrod) cohomology]]).

\end{prop}
\begin{proof}
The [[mapping cone]] of any map out of a [[CW-complex]] represents the [[homotopy cofiber]] of that map (\href{Introduction+to+Stable+homotopy+theory+--+P#StandardTopologicalMappingConeIsHomotopyCofiber}{exmpl.}). Moreover, transformation by (weak) homotopy equivalences between morphisms induces a (weak) homotopy equivalence on their homotopy fibers (\href{Introduction+to+Stable+homotopy+theory+--+P#FiberOfFibrationIsCompatibleWithWeakEquivalences}{prop.}). But we have such a weak homotopy equivalence, given by contracting away the fibers of the vector bundle:

\begin{displaymath}
\itexarray{
    V\setminus X &\longrightarrow& V
    \\
    {}^{\mathllap{\in W_{cl}}}\downarrow && \downarrow^{\mathrlap{\in W_{cl}}}
    \\
    S(V) &\hookrightarrow& D(V)
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{behaviour_under_direct_sum_of_vector_bundles}{}\subsubsection*{{Behaviour under direct sum of vector bundles}}\label{behaviour_under_direct_sum_of_vector_bundles}

\begin{prop}
\label{ThomSpaceOfDirectSumAsQuotientOfThomSpacesOfPullbacks}\hypertarget{ThomSpaceOfDirectSumAsQuotientOfThomSpacesOfPullbacks}{}
Let $V_1,V_2 \to X$ be two real [[vector bundles]]. Then the Thom space (def. \ref{ThomSpace}) of the [[direct sum of vector bundles]] $V_1 \oplus V_2 \to X$ is expressed in terms of the Thom space of the [[pullback bundles]] $V_2|_{D(V_1)}$ and $V_2|_{S(V_1)}$ of $V_2$ to the disk/sphere bundle of $V_1$ as

\begin{displaymath}
Th(V_1 \oplus V_2)
  \simeq
  Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Notice that

\begin{enumerate}%
\item $D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1)$;


\item $S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)})$.



\end{enumerate}
(Since a point at radius $r$ in $V_1 \oplus V_2$ is a point of radius $r_1 \leq r$ in $V_2$ and a point of radius $\sqrt{r^2 - r_1^2}$ in $V_1$.)

\end{proof}
\begin{prop}
\label{SuspensionOfThomSpaces}\hypertarget{SuspensionOfThomSpaces}{}
For $V$ a [[vector bundle]] then the Thom space (def. \ref{ThomSpace}) of $\mathbb{R}^n \oplus V$, the [[direct sum of vector bundles]] with the trivial rank $n$ vector bundle, is [[homeomorphism|homeomorphic]] to the [[smash product]] of the Thom space of $V$ with the $n$-[[sphere]] (the $n$-fold [[reduced suspension]]).

\begin{displaymath}
Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Apply prop. \ref{ThomSpaceOfDirectSumAsQuotientOfThomSpacesOfPullbacks} with $V_1 = \mathbb{R}^n$ and $V_2 = V$. Since $V_1$ is a trivial bundle, then

\begin{displaymath}
V_2|_{D(V_1)} \simeq V_2\times D^n
\end{displaymath}
(as a bundle over $X\times D^n$) and similarly

\begin{displaymath}
V_2|_{S(V_1)} \simeq V_2\times S^n
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{SuspensionPropertyOfThomSpacesInducesThomSpectra}\hypertarget{SuspensionPropertyOfThomSpacesInducesThomSpectra}{}
Prop. \ref{SuspensionOfThomSpaces} implies that for every vector bundle $V$ the sequence of spaces $Th(\mathbb{R}^n \oplus V)$ forms a [[suspension spectrum]]: this is \emph{the [[Thom spectrum]]} of $V$.

\end{remark}
\begin{example}
\label{ThomSpaceConstructionReducingToSuspension}\hypertarget{ThomSpaceConstructionReducingToSuspension}{}
By prop. \ref{SuspensionOfThomSpaces} and remark \ref{ThomSpaceForRankZeroBundle} the Thom space (def. \ref{ThomSpace}) of a trivial vector bundle of rank $n$ is the $n$-fold [[suspension]] of the base space

\begin{displaymath}
\begin{aligned}
    Th(X \times \mathbb{R}^n) 
    & \simeq S^n \wedge Th(X\times \mathbb{R}^0) 
    \\
    & \simeq S^n \wedge (X_+)
  \end{aligned}
  \,.
\end{displaymath}
Therefore a general Thom space may be thought of as a ``twisted [[reduced suspension]]'', with twist encoded by a vector bundle (or rather by its underlying [[spherical fibration]]). See at \emph{\href{Thom+spectrum#ForInfinityModuleBundles}{Thom spectrum -- For infinity-module bundles}} for more on this.

Correspondingly the \emph{[[Thom isomorphism]]} for a given Thom space is a twisted version of the \emph{[[suspension isomorphism]]}.

\end{example}
\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}
\begin{remark}
\label{}\hypertarget{}{}
Prop. \ref{ThomSpaceOfExternalProductOfVectorBundles} induces on the [[Thom spectra]] of remark \ref{SuspensionPropertyOfThomSpacesInducesThomSpectra} the structure of [[ring spectra]].

\end{remark}
\hypertarget{cwstructure}{}\subsubsection*{{CW-structure}}\label{cwstructure}

If the base space of the vector bundle carries the structure of a [[CW-complex]], then its Thom space (def. \ref{ThomSpace}) canonically inherits the structure of a CW-complex, too:

\begin{lemma}
\label{ThomSpaceCWStructure}\hypertarget{ThomSpaceCWStructure}{}
Let $V \to X$ be a [[vector bundle]] of [[rank]] $n \geq 1$. over a [[CW-complex]] $X$.

Then $Th(V)$ has the structure of a [[CW-complex]] with

\begin{enumerate}%
\item $S(E)/S(E)$ the only 0-cell


\item precisely one $(n+k)$-cell $D^{k+n}\to Th(V)$ for each $k$-cell $D^k \to X$ of $X$, given as the [[pullback]]

\begin{displaymath}
\itexarray{
    D^{k+n} &\longrightarrow& D(V) &\longrightarrow& D(V)/S(V) = Th(V)
    \\
    \downarrow &(pb)& \downarrow
    \\
    D^k &\longrightarrow& X
  }
  \,.
\end{displaymath}


\end{enumerate}
\end{lemma}
(e.g. \hyperlink{Cruz04}{Cruz 04, lemma 6})

In particular, $Th(V)$ has a single $n$-cell and an $(n+1)$-cell for each 1-cell of $X$. There are no cells in $Th(C)$ between dimension $0$ and $n$. The cellular boundary of an $(n+1)$-cell is 0 if $V$ is orientable over the corresponding 1-cell of $X$, and it is twice the $n$-cell in the opposite case. Thus $H^n(Th(V);\mathbb{Z})$ is $\mathbb Z$ if $V$ is orientable and $0$ if $V$ is non-orientable. In the orientable case a generator of $H^n(Th(V);{\mathbb Z})$ restricts to a generator of $H^n(S^n;\mathbb{Z})$ in the ``fiber'' $S^n$ of $Th(V)$ over the 0-cell of $X$, hence the same is true for all the ``fibers'' $S^n$ and so one has a [[Thom class]].

(\href{http://mathoverflow.net/a/107864/381}{MO discussion})

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

\begin{remark}
\label{OrdinaryCohomologyOfThomSpaceInLowDegree}\hypertarget{OrdinaryCohomologyOfThomSpaceInLowDegree}{}
Given a [[vector bundle]] $V \to X$ of [[rank]] $n$, then the [[reduced cohomology|reduced]] [[ordinary cohomology]] of its [[Thom space]] $Th(V)$ (def. \ref{ThomSpace}) vanishes in degrees $\lt n$:

\begin{displaymath}
\tilde H^{\bullet \lt n}(Th(V))
  \simeq
  H^{\bullet \lt n}(D(V), S(V))
  \simeq 0
  \,.
\end{displaymath}
\end{remark}
\begin{proof}
Consider the [[long exact sequence]] of [[relative cohomology]] (\href{generalized+cohomology#ExactnessUnreduced}{here})

\begin{displaymath}
\cdots
    \to
  H^{\bullet-1}(D(V))
   \overset{i^\ast}{\longrightarrow}
  H^{\bullet-1}(S(V))
   \longrightarrow
  H^\bullet(D(V), S(V))
    \longrightarrow
  H^{\bullet}(D(V))
   \overset{i^\ast}{\longrightarrow}
  H^{\bullet}(S(V))
    \to
  \cdots  
  \,.
\end{displaymath}
Since the cohomology in degree $k$ only depends on the $k$-skeleton, and since for $k \lt n$ the $k$-skeleton of $S(V)$ equals that of $X$, and since $D(V)$ is even homotopy equivalent to $X$, the morhism $i^\ast$ is an isomorphism in degrees lower than $n$. Hence by exactness of the sequence it follows that $H^{\bullet \lt n}(D(V),S(V)) = 0$.

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

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


\item [[one-point compactification]]


\item [[Thom spectrum]]


\item [[Thom isomorphism]]


\item [[projective bundle]]



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

The [[Thom isomorphism]] for Thom spaces was originally found in

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

\end{itemize}
For general discussion see

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Thom complexes}, Proc. London Math. Soc. \textbf{11} (1961) pp. 291--310


\item [[Yuli Rudyak]], \emph{On Thom spectra, orientability, and cobordism}, Springer 1998 \href{http://books.google.hr/books?isbn=3540620435}{googB}


\item [[eom]], \emph{\href{http://eom.springer.de/t/t092680.htm}{Thom space}}


\item [[Dale Husemöller]], \emph{Fibre bundles} , McGraw-Hill (1966)


\item myyn.org \href{http://myyn.org/m/article/thom-space}{Thom space}, \href{http://myyn.org/m/article/thom-class}{Thom class}, \href{http://myyn.org/m/article/thom-isomorphism-theorem}{Thom isomorphism theorem}



\end{itemize}
See also

\begin{itemize}%
\item [[Robert Stong]], \emph{Notes on cobordism theory} , Princeton Univ. Press (1968)


\item W.B. Browder, \emph{Surgery on simply-connected manifolds} , Springer (1972)


\item Martin Vito Cruz, \emph{An introduction to cobordism}, 2004 (\href{https://math.berkeley.edu/~hutching/teach/215b-2004/vitocruz.pdf}{pdf})



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



\end{document}