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\section*{Euler class}

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{CupSquare}{Cup square}\dotfill \pageref*{CupSquare} \linebreak
\noindent\hyperlink{WhitneySumFormula}{Whitney sum formula}\dotfill \pageref*{WhitneySumFormula} \linebreak
\noindent\hyperlink{poincarhopf_theorem}{Poincaré–Hopf theorem}\dotfill \pageref*{poincarhopf_theorem} \linebreak
\noindent\hyperlink{on_unit_sphere_bundles}{On unit sphere bundles}\dotfill \pageref*{on_unit_sphere_bundles} \linebreak
\noindent\hyperlink{fiber_integration}{Fiber integration}\dotfill \pageref*{fiber_integration} \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{euler_forms}{Euler forms}\dotfill \pageref*{euler_forms} \linebreak


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

The \emph{Euler class} $\chi$ (or $e$) is a [[characteristic class]] of the [[special orthogonal group]], hence of [[oriented]] [[real vector bundles]].

The Euler class of the [[tangent bundle]] of a [[smooth manifold]] $X$, evaluated on its [[fundamental class]], is its [[Euler characteristic]] $\chi[X]$.

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

\hypertarget{CupSquare}{}\subsubsection*{{Cup square}}\label{CupSquare}

For $E$ a [[vector bundle]] of [[even number|even]] [[rank of a vector bundle|rank]] $rank(E) = 2 k$, the [[cup product]] of the [[Euler class]] with itself equals the $k$th [[Pontryagin class]]

\begin{equation}
\chi(E) \smile \chi(E)
  \;=\;
  p_k(E)
  \,.
\label{EulerSquareIsPontryagin}\end{equation}
(e.g. \hyperlink{Walschap04}{Walschap 04, Section 6.3, p. 187})

When the Euler class is represented by the [[Euler form]] of a [[connection]] $\nabla$ on $E$, which then is [[fiber]]-wise proportional to the [[Pfaffian]] of the [[curvature form]] $F_\nabla$ of $\nabla$, the relation \eqref{EulerSquareIsPontryagin} corresponds to the fact that the product of a [[Pfaffian]] with itself is the [[determinant]]: $\big( Pf(F_\nabla) \big)^2 = det(F_\nabla)$.

\hypertarget{WhitneySumFormula}{}\subsubsection*{{Whitney sum formula}}\label{WhitneySumFormula}

\begin{prop}
\label{EulerClassOfWhitneySumIsCupProductOfEulerClasses}\hypertarget{EulerClassOfWhitneySumIsCupProductOfEulerClasses}{}
\textbf{([[Euler class]] takes [[Whitney sum]] to [[cup product]])}

The Euler class of the [[Whitney sum]] of two [[orthogonal group|oriented]] [[real vector bundles]] to the [[cup product]] of the separate Euler classes:

\begin{displaymath}
\chi( E \oplus F )
  \;=\;
  \chi(E)  \smile \chi(F)
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Walschap04}{Walschap 04, Section 6.4})

\hypertarget{poincarhopf_theorem}{}\subsubsection*{{Poincaré–Hopf theorem}}\label{poincarhopf_theorem}

\begin{itemize}%
\item [[Poincaré–Hopf theorem]]

\end{itemize}
\hypertarget{on_unit_sphere_bundles}{}\subsubsection*{{On unit sphere bundles}}\label{on_unit_sphere_bundles}

\begin{prop}
\label{TrivializationOfEulerFormOnUnitSphereBundle}\hypertarget{TrivializationOfEulerFormOnUnitSphereBundle}{}
Let $X$ be a [[smooth manifold]] and $E \overset{\pi}{\longrightarrow} X$ an oriented [[real vector bundle]] of [[even number|even]] [[rank of a vector bundle|rank]], $rank(E) = 2k + 2$.

For any choice of [[connection on a bundle|connection]] $\nabla$ on $E$ ($SO(dim(X))$-connection), let $\chi(\nabla_E) \in \Omega^{2k}(X)$ denote the corresponding [[Euler form]].

Then the [[pullback of differential forms|pullback]] of the [[Euler form]] $\chi(\nabla_E)$ to the [[unit sphere bundle]] $S(E) \overset{S(\pi)}{\longrightarrow} X$ is [[exact differential form|exact]]

\begin{displaymath}
\big( S(\pi) \big)^\ast
  \chi(\nabla_E)
  \;=\;
  d \Omega
\end{displaymath}
such that the trivializing form has (minus) unit [[integration of differential forms|integral]] over any of the [[n-sphere|(2k+1)-sphere]]-[[fibers]] $S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E)$:

\begin{equation}
\int_{S^{2k+1}} \iota_x^\ast \Omega 
  \;=\;
  -1
  \,.
\label{FiberIntegrationOfTrivialization}\end{equation}
\end{prop}
(e.g. \hyperlink{Walschap04}{Walschap 04, Chapter 6.6, Thm. 6.1, p. 201-202}, \hyperlink{Poor07}{Poor 07, 3.68}, \hyperlink{Nie09}{Nie 09})

\hypertarget{fiber_integration}{}\subsubsection*{{Fiber integration}}\label{fiber_integration}

\begin{prop}
\label{FiberIntegrationOfCupPowersOfChiOver4Sphere}\hypertarget{FiberIntegrationOfCupPowersOfChiOver4Sphere}{}
Let

\begin{displaymath}
\itexarray{
    S^4
    &\longrightarrow& 
    B Spin(4)
    \\
    &&
    \big\downarrow^{\mathrlap{\pi}}
    \\
    && 
    B Spin(5)
  }
\end{displaymath}
be the [[spherical fibration]] of [[classifying spaces]] induced from the canonical inclusion of [[Spin(4)]] into [[Spin(5)]] and using that the [[4-sphere]] is equivalently the [[coset space]] $S^4 \simeq Spin(5)/Spin(4)$ (\href{sphere#nSphereAsCosetSpace}{this Prop.}).

Then the [[fiber integration]] of the odd [[cup product|cup powers]] $\chi^{2k+1}$ of the [[Euler class]] $\chi \in H^4\big( B Spin(4), \mathbb{Z}\big)$ (see \href{Spin4#IntegralCohomologyOfClassifyingSpace}{this Prop}) are proportional to [[cup product|cup powers]] of the [[second Pontryagin class]]

\begin{displaymath}
\pi_\ast
  \left(
    \chi^{2k+1}
  \right)
  \;=\;
  2 \big( p_2 \big)^k 
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big) 
  \,,
\end{displaymath}
for instance

\begin{displaymath}
\begin{aligned}
    \pi_\ast
    \big(
      \chi
    \big)
    & = 
    2 
    \\
    \pi_\ast
    \left(
      \chi^3
    \right)
    & = 
    2 p_2 
    \\
    \pi_\ast
    \left(
      \chi^5
    \right)
    & = 
    2 (p_2)^2 
  \end{aligned}
    \;\;\in\;\;
    H^4\big( B Spin(5), \mathbb{Z} \big) 
  \,;
\end{displaymath}
while the [[fiber integration]] of the even [[cup product|cup powers]] $\chi^{2k}$ vanishes

\begin{displaymath}
\pi_\ast
  \left(
    \chi^{2k}
  \right)
  \;=\;
  0
  \;\;\in\;\;
  H^4\big( B Spin(5), \mathbb{Z} \big) 
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BottCattaneo98}{Bott-Cattaneo 98, Lemma 2.1})

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

\begin{itemize}%
\item [[Pontryagin class]], [[Stiefel-Whitney class]], [[Wu class]]


\item [[I8|one-loop anomaly polynomial]]



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

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

\begin{itemize}%
\item [[Raoul Bott]], [[Loring Tu]], Chapter 11 of \emph{[[Differential Forms in Algebraic Topology]]}, Graduate Texts in Mathematics 82, Springer 1982 (\href{https://doi.org/10.1007/BFb0063500}{doi:10.1007/BFb0063500})


\item [[Allen Hatcher]], \emph{Euler and Pontryagin classes}, section 3.2 in \emph{\href{http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html}{Vector bundles and K-theory}} (\href{http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf}{pdf})


\item Anant R. Shastri, section 2.1 of \emph{Vector bundles and Characteristic Classes} (\href{http://www.math.iitb.ac.in/~ars/seminar/bundle.pdf}{pdf})


\item Michael Hutchings, section 5 of \emph{Cup product and intersections} (\href{https://math.berkeley.edu/~hutching/teach/215b-2011/cup.pdf}{pdf})


\item Gerard Walschap, chapter 6.3 of \emph{Metric Structures in Differential Geometry}, Graduate Texts in Mathematics, Springer 2004



\end{itemize}
Discussion of [[fiber integration]]:

\begin{itemize}%
\item [[Raoul Bott]], [[Alberto Cattaneo]], \emph{Integral Invariants of 3-Manifolds}, J. Diff. Geom., 48 (1998) 91-133 (\href{https://arxiv.org/abs/dg-ga/9710001}{arXiv:dg-ga/9710001})

\end{itemize}
Discussion for [[projective modules]]

\begin{itemize}%
\item Satya Manda, \emph{An overview of Euler class theory} (\href{http://mandal.faculty.ku.edu/talks/amsTalk06.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia \href{http://en.wikipedia.org/wiki/Euler_class}{Euler class}


\item Robert F. Brown, \emph{On the Lefschetz number and the Euler class}, Transactions of the AMS \textbf{118}, (1965) (\href{http://www.jstor.org/pss/1993952}{JSTOR})


\item Solomon Jekel, \emph{A simplicial formula and bound for the Euler class}, Israel Journal of Mathematics \textbf{66}, n. 1-3, 247-259 (1989)



\end{itemize}
\hypertarget{euler_forms}{}\subsubsection*{{Euler forms}}\label{euler_forms}

Discussion of \emph{[[Euler forms]]} ([[differential form]]-representatives of Euler classes in [[de Rham cohomology]]) as [[Pfaffians]] of [[curvature forms]]:

\begin{itemize}%
\item [[Shiing-Shen Chern]], \emph{A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds}, Annals of Mathematics, Second Series, Vol. 45, No. 4 (1944), pp. 747-752 (\href{https://www.jstor.org/stable/1969302}{jstor:1969302})


\item [[Raoul Bott]], [[Loring Tu]], Chapter 11 of \emph{[[Differential Forms in Algebraic Topology]]}, Graduate Texts in Mathematics 82, Springer 1982 (\href{https://doi.org/10.1007/BFb0063500}{doi:10.1007/BFb0063500})


\item [[Varghese Mathai]], [[Daniel Quillen]], below (7.3) of \emph{Superconnections, Thom classes, and equivariant differential forms}, Topology Volume 25, Issue 1, 1986 ()


\item Siye Wu, Section 2.2 of \emph{Mathai-Quillen Formalism}, pages 390-399 in \emph{Encyclopedia of Mathematical Physics} 2006 (\href{https://arxiv.org/abs/hep-th/0505003}{arXiv:hep-th/0505003})


\item \hyperlink{Walschap04}{Walschap 04, section 6.3}


\item Walter A. Poor, 3.58 of \emph{Differential Geometric Structures}, Dover Books on Mathematics, 2007


\item Zhaohu Nie, \emph{Secondary Chern-Euler forms and the Law of Vector Fields} (\href{https://arxiv.org/abs/0909.4754}{arXiv:0909.4754}


\item [[Hiro Lee Tanaka]], \emph{Pfaffians and the Euler class}, 2014 (\href{http://www.hiroleetanaka.com/pdfs/2014-fall-230a-lecture-26-gauss-bonnet-chern.pdf}{pdf})


\item [[Liviu Nicolaescu]], Section 8.3.2 of \emph{Lectures on the Geometry of Manifolds}, 2018 (\href{https://www3.nd.edu/~lnicolae/Lectures.pdf}{pdf}, \href{https://mathoverflow.net/a/117982/381}{MO comment})



\end{itemize}
[[!redirects Euler classes]]

[[!redirects Euler form]] [[!redirects Euler forms]]



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