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\section*{Thom isomorphism}

\begin{quote}%
This entry is about the isomorphisms in [[cohomology]] induced by [[Thom classes]]. For the [[Pontrjagin-Thom isomorphism]] in [[cobordism theory]] see at \emph{[[Thom's theorem]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{concretely}{Concretely}\dotfill \pageref*{concretely} \linebreak
\noindent\hyperlink{in_ordinary_cohomology}{In ordinary cohomology}\dotfill \pageref*{in_ordinary_cohomology} \linebreak
\noindent\hyperlink{in_generalized_cohomology}{In generalized cohomology}\dotfill \pageref*{in_generalized_cohomology} \linebreak
\noindent\hyperlink{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \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}

For $H$ being [[ordinary cohomology]] with [[coefficients]] in a [[ring]], and $V \to X$ a [[vector bundle]] of rank $n$ over a simply connected CW-complex, the \textbf{Thom isomorphism} is the morphism

\begin{displaymath}
c \cup (-)
    \;\colon\; 
  H^\bullet(X) 
     \stackrel{\simeq}{\longrightarrow} 
  \tilde H^{\bullet + n}(Th(V))
  \,.
\end{displaymath}
from the [[cohomology]] of $X$ to the [[reduced cohomology]] of the [[Thom space]] $Th(V)$, given by pullback to the [[Thom space]] followed by [[cup product]] with a [[Thom class]] $c \in H^n(Th(V))$. That this is indeed an [[isomorphism]] follows via the [[Leray-Hirsch theorem]] (see e.g. \hyperlink{Ebert12}{Ebert 12, 2.3,2.4}) or from running a [[Serre spectral sequence]] (e.g. \hyperlink{Kochman96}{Kochman 96, section 2.6}).

In the special case that the vector bundle is trivial of rank $n$, then its [[Thom space]] coincides with the $n$-fold [[suspension]] of the base space (\href{Thom+space#ThomSpaceConstructionReducingToSuspension}{exmpl.}) and the Thom isomorphism coincides with the [[suspension isomorphism]]. In this sense the Thom isomorphism may be regarded as a \emph{twisted suspension isomorphism}.

More generally for $E$ a [[multiplicative cohomology theory]], and $V \to X$ a [[vector bundle]] of rank $n$, which is $E$-[[orientation in generalized cohomology|orientable]], there is a generalization to a \textbf{Thom-Dold isomorphism}

\begin{displaymath}
c \cup (-)
    \;\colon\; 
  E^\bullet(X) 
     \stackrel{\simeq}{\longrightarrow} 
  \tilde E^{\bullet + n}(Th(V))
\end{displaymath}
(e.g. \hyperlink{Rudyak98}{Rudyak 98, chapter V, 1.3}, \hyperlink{Kochman96}{Kochman 96, prop. 4.3.6})

One may think of the Thom isomorphism from left to right as [[cup product|cupping]] with a generalized [[volume form]] on the fibers, and from right to left as performing [[fiber integration]] against this volume form.

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

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

\hypertarget{in_ordinary_cohomology}{}\paragraph*{{In ordinary cohomology}}\label{in_ordinary_cohomology}

\begin{prop}
\label{Smooth0TypeIsSheavesOnSmoothMfd}\hypertarget{Smooth0TypeIsSheavesOnSmoothMfd}{}
Let $V \to B$ be a topological [[vector bundle]] of [[rank]] $n \gt 0$ over a [[simply connected topological space|simply connected]] [[CW-complex]] $B$. Let $R$ be a [[commutative ring]].

There exists an element $c \in H^n(Th(V);R)$ (in the [[ordinary cohomology]], with [[coefficients]] in $R$, of the [[Thom space]] of $V$, called a \textbf{[[Thom class]]}) such that forming the [[cup product]] with $c$ induces an [[isomorphism]]

\begin{displaymath}
H^\bullet(B;R)
    \overset{c \cup (-)}{\longrightarrow}
  \tilde H^{\bullet + n}(Th(V);R)
\end{displaymath}
of degree $n$ from the unreduced [[cohomology group]] of $B$ to the [[reduced cohomology]] of the [[Thom space]] of $V$.

\end{prop}
\begin{proof}
\textbf{(of Thom isomorphism via fiberwise Thom spaces)}

Choose an [[orthogonal structure]] on $V$. Consider the \emph{fiberwise} [[cofiber]]

\begin{displaymath}
E \coloneqq D(V)/_B S(V)
\end{displaymath}
of the inclusion of the [[unit sphere]] bundle into the unit disk bundle of $V$ (\href{Thom+space#ThomSpace}{def.}).

\begin{displaymath}
\itexarray{
    S^{n-1} &\hookrightarrow& D^n &\longrightarrow& S^n
    \\
    \downarrow && \downarrow && \downarrow
    \\
    S(V) &\hookrightarrow& D(V) &\longrightarrow& E
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{p}}
    \\
    B &=& B &=& B
  }
\end{displaymath}
Observe that this has the following properties

\begin{enumerate}%
\item $E \overset{p}{\to} B$ is an [[n-sphere]] [[fiber bundle]], hence in particular a [[Serre fibration]];


\item the [[Thom space]] $Th(V)\simeq E/B$ is the quotient of $E$ by the base space, because of the [[pasting law]] applied to the following pasting diagram of [[pushout]] squares

\begin{displaymath}
\itexarray{ 
    S(V) &\longrightarrow& D(V)
    \\
    \downarrow &(po)& \downarrow
    \\
    B &\longrightarrow& D(V)/_B S(V)
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& Th(V)
  }
\end{displaymath}

\item hence the [[reduced cohomology]] of the Thom space is (\href{Introduction+to+Stable+homotopy+theory+--+S#ReducedToUnreducedGeneralizedCohomology}{def.}) the [[relative cohomology]] of $E$ relative $B$

\begin{displaymath}
\tilde H^\bullet(Th(V);R) \simeq H^\bullet(E,B;R)
  \,.
\end{displaymath}

\item $E \overset{p}{\to} B$ has a global [[section]] $B \overset{s}{\to} E$ (given over any point $b \in B$ by the class of any point in the fiber of $S(V) \to B$ over $b$; or abstractly: induced via the above pushout by the commutation of the projections from $D(V)$ and from $S(V)$, respectively).



\end{enumerate}
In the following we write $H^\bullet(-)\coloneqq H^\bullet(-;R)$, for short.

By the first point, there is the [[Thom-Gysin sequence]], an [[exact sequence]] running vertically in the following diagram

\begin{displaymath}
\itexarray{
    && H^\bullet(B)
    \\
    && {}^{\mathllap{p^\ast}}\downarrow  & \searrow^{\mathrlap{\simeq}}
    \\
    \tilde H^\bullet(Th(V))
      &\longrightarrow&
    H^\bullet(E)
      &\underset{s^\ast}{\longrightarrow}&
    H^\bullet(B)
    \\
    && \downarrow
    \\
    && H^{\bullet-n}(B)
  }
  \,.
\end{displaymath}
By the second point above this is [[split exact sequence|split]], as shown by the diagonal isomorphism in the top right. By the third point above there is the horizontal exact sequence, as shown, which is the \href{generalized+%28Eilenberg-Steenrod%29+cohomology#ExactnessUnreduced}{exact sequence in relative cohomology} $\cdots \to H^\bullet(E,B) \to H^\bullet(E) \to H^\bullet(B) \to \cdots$ induced from the section $B \hookrightarrow E$.

Hence using the splitting to decompose the term in the middle as a [[direct sum]], and then using horizontal and vertical exactness at that term yields

\begin{displaymath}
\itexarray{
    && H^\bullet(B)
    \\
    && {}^{\mathllap{(0,id)}}\downarrow  & \searrow^{\mathrlap{\simeq}}
    \\
    \tilde H^\bullet(Th(V))
      &\overset{(id,0)}{\hookrightarrow}&
    \tilde H^\bullet(Th(V))
      \oplus 
    H^\bullet(B)
      &\underset{(0,id)}{\longrightarrow}&
    H^\bullet(B)
    \\
    && \downarrow^{\mathrlap{(id,0)}}
    \\
    && H^{\bullet-n}(B)
  }
\end{displaymath}
and hence an isomorphism

\begin{displaymath}
\tilde H^\bullet(Th(V))
    \overset{\simeq}{\longrightarrow}
  H^{\bullet-n}(B)
  \,.
\end{displaymath}
To see that this is the inverse of a morphism of the form $c \cup (-)$, inspect the \href{Thom-Gysin+sequence#ProofOfThomGysinSequence}{proof of the Gysin sequence}. This shows that $H^{\bullet-n}(B)$ here is identified with elements that on the second page of the corresponding [[Serre spectral sequence]] are cup products

\begin{displaymath}
\iota \cup b
\end{displaymath}
with $\iota$ fiberwise the canonical class $1 \in H^n(S^n)$ and with $b \in H^\bullet(B)$ any element. Since $H^\bullet(-;R)$ is a [[multiplicative cohomology theory]] (because the [[coefficients]] form a [[ring]] $R$), cup producs are preserved as one passes to the $E_\infty$-page of the spectral sequence, and the morphism $H^\bullet(E) \to B^\bullet(B)$ above, hence also the isomorphism $\tilde H^\bullet(Th(V)) \to H^\bullet(B)$, factors through the $E_\infty$-page (see towards the end of the \href{Thom-Gysin+sequence#ProofOfThomGysinSequence}{proof of the Gysin sequence}). Hence the image of $\iota$ on the $E_\infty$-page is the Thom class in question.

\end{proof}
\hypertarget{in_generalized_cohomology}{}\paragraph*{{In generalized cohomology}}\label{in_generalized_cohomology}

Let $E$ be a [[generalized (Eilenberg-Steenrod) cohomology]] theory. First observe that an [[E-orientation]] on $V \to X$ induces an $H \pi_0(E)$-orientation, i.e. in [[ordinary cohomology]] with coefficients in the degree-0 ground ring.

To see this, let's assume $E$ is [[connective spectrum|connective]]. Consider the [[relative Atiyah-Hirzebruch spectral sequence]]

\begin{displaymath}
\tilde H^p(Th(V), E^q(\ast))
  \simeq
  H^p(D(V),S(V), E^q(\ast))
    \;\Rightarrow\;
  E^\bullet(D(V), S(V))
  \simeq
  \tilde E^\bullet(Th(V))
\end{displaymath}
Since $(D(V), S(V))$ is $(n-1)$-connected for a rank $n$ vector bundle, then $E_2^{p \lt n, q} = 0$. Hence the edge homomorphism

\begin{displaymath}
\tilde H^k(Th(V), E^0(\ast)) \longrightarrow \tilde H^0(Th(V))
\end{displaymath}
is an [[isomorphism]], and one checks that it sends Thom classes to Thom classes.

For a fully detailed account see (\hyperlink{Pedrotti16}{Pedrotti 16}).

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

A general abstract discussion is around page 30, 31 of (\hyperlink{ABGHR}{ABGHR}).

(\ldots{})

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The Thom isomorphism is used to define [[fiber integration]] of multiplicative cohomology theories.

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

\begin{itemize}%
\item [[Thom-Gysin sequence]]


\item [[cobordism theory]]


\item [[Thom space]], [[Thom spectrum]]


\item [[fiber integration]]


\item [[Pontrjagin-Thom collapse map]]



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

The original proof that the Thom isomorphism is indeed an [[isomorphism]] 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} Comm. Math. Helv. , 28 (1954) pp. 17--86

\end{itemize}
The argument via a [[Serre spectral sequence]] for a relative fibration seems to be due to

\begin{itemize}%
\item W. H. Cockcroft, \emph{On the Thom isomorphism Theorem}, Mathematical Proceedings of the Cambridge Philosophical Society (1962), 58 (\href{http://journals.cambridge.org/abstract_S0305004100036409}{web}, \href{http://journals.cambridge.org/action/displayFulltext?type=1&fid=2056796&jid=PSP&volumeId=58&issueId=02&aid=2056788&bodyId=&membershipNumber=&societyETOCSession=}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Stanley Kochman]], section 2.6 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


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



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Akhil Mathew]], \emph{\href{https://amathew.wordpress.com/2010/12/11/the-thom-isomorphism-theorem/}{The Thom isomorphism theorem}}, 2010


\item [[Johannes Ebert]], sections 2.3, 2.4 of \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 8 of \emph{Bordism: old and new}, 2013 (\href{http://www.ma.utexas.edu/users/dafr/bordism.pdf}{pdf})


\item [[Riccardo Pedrotti]], \emph{Complex oriented cohomology, generalized orientation and Thom isomorphism}, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])



\end{itemize}
A discussion in [[differential geometry]] with fiberwise compactly supported differential forms is around theorem 6.17 of

\begin{itemize}%
\item [[Raoul Bott]], [[Loring Tu]], \emph{[[Differential Forms in Algebraic Topology]]}

\end{itemize}
A comprehensive general abstract account for [[multiplicative cohomology theories]] in terms of [[E-infinity ring]] [[spectra]] 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})

\end{itemize}
An alternative simple formulation in terms of geometric cycles as in [[bivariant cohomology theory]] is in

\begin{itemize}%
\item Martin Jakob, \emph{A note on the Thom isomorphism in geometric (co)homology} (\href{http://arxiv.org/abs/math/0403540}{arXiv:math/0403540})

\end{itemize}
See also

\begin{itemize}%
\item [[Albrecht Dold]], \emph{Relations between ordinary and extraordinary homology} , Colloq. Algebraic Topology, August 1--10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2--9


\item [[Yuli Rudyak]], \emph{On the Thom--Dold isomorphism for nonorientable bundles} Soviet Math. Dokl. , 22 (1980) pp. 842--844 Dokl. Akad. Nauk. SSSR , 255 : 6 (1980) pp. 1323--1325


\item [[Robert Switzer]], \emph{Algebraic topology - homotopy and homology} , Springer (1975)


\item myyn.org (Planetmath) \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}
Formalization in [[homotopy type theory]] is discussed in

\begin{itemize}%
\item [[Guillaume Brunerie]], \emph{On the homotopy groups of spheres in homotopy type theory} (\href{http://arxiv.org/abs/1606.05916}{arXiv:1606.05916})

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

[[!redirects Thom-Dold isomorphism]] [[!redirects Thom-Dold isomorphisms]]



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