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\begin{document}

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\section*{Chern 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{existence}{Existence}\dotfill \pageref*{existence} \linebreak
\noindent\hyperlink{first_chern_class}{First Chern class}\dotfill \pageref*{first_chern_class} \linebreak
\noindent\hyperlink{SplittingPrinciple}{Splitting principle and Chern roots}\dotfill \pageref*{SplittingPrinciple} \linebreak
\noindent\hyperlink{WhitneySumFormula}{Whitney sum formula}\dotfill \pageref*{WhitneySumFormula} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ChernClassesOfLinearRepresentations}{Chern classes of linear representations}\dotfill \pageref*{ChernClassesOfLinearRepresentations} \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 ordinary \emph{Chern classes} are the [[integral cohomology|integral]] [[characteristic classes]]

\begin{displaymath}
c_i : B U \to B^{2 i} \mathbb{Z}
\end{displaymath}
of the [[classifying space]] $B U$ of the [[unitary group]].

Accordingly these are characteristic classes in [[ordinary cohomology]] of [[unitary group|U]]-[[principal bundles]] and hence of [[complex vector bundle]]

The first Chern class is the unique characteristic class of [[circle group]]-principal bundles.

The analogous classes for the [[orthogonal group]] are the [[Pontryagin classes]].

More generally, there are \emph{[[generalized Chern classes]]} for any [[complex oriented cohomology theory]] (\hyperlink{Adams74}{Adams 74}, \hyperlink{Lurie10}{Lurie 10}).

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

\begin{defn}
\label{}\hypertarget{}{}
For $n \geq 1$ the \emph{[[universal characteristic classes|universal]] Chern [[characteristic classes|classes]]}

\begin{displaymath}
c_i 
    \;\in\; 
  H^{2i}
  \big(
    B U(n), \mathbb{Z}
  \big)
\end{displaymath}
of the [[classifying space]] $B U(n)$ of the [[unitary group]] are the [[cohomology classes]] of $B U(n)$ in [[integral cohomology]] that are characterized as follows:

\begin{enumerate}%
\item $c_0 = 1$ and $c_i = 0$ if $i \gt n$;


\item for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;


\item under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;


\item under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.



\end{enumerate}
The corresponding \emph{total Chern class} is the formal sum

\begin{displaymath}
c 
  \;\coloneqq\;
  1 + c_1 + c+2 + \cdots
  \;\in\;
  \underset{k}{\prod} 
  H^{2k}
  \big(
    B U(n)
  \big)
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{existence}{}\subsubsection*{{Existence}}\label{existence}

\begin{prop}
\label{GeneratorsOfCohomologyOfBunChernClasses}\hypertarget{GeneratorsOfCohomologyOfBunChernClasses}{}
The [[cohomology ring]] of the [[classifying space]] $B U(n)$ (for the [[unitary group]] $U(n)$) is the [[polynomial ring]] on generators $\{c_k\}_{k = 1}^{n}$ of degree 2, called the \emph{Chern classes}

\begin{displaymath}
H^\bullet(B U(n), \mathbb{Z})
    \simeq
  \mathbb{Z}[c_1, \cdots, c_n]
  \,.
\end{displaymath}
Moreover, for $B i \colon B U(n_1) \longrightarrow BU(n_2)$ the canonical inclusion for $n_1 \leq n_2 \in \mathbb{N}$, then the induced pullback map on cohomology

\begin{displaymath}
(B i)^\ast 
     \;\colon\;
   H^\bullet(B U(n_2))
     \longrightarrow
   H^\bullet(B U(n_1))
\end{displaymath}
is given by

\begin{displaymath}
(B i)^\ast(c_k)
  \;=\;
  \left\{
    \itexarray{
      c_k & for \; 1 \leq k \leq n_1
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, theorem 2.3.1})

\begin{proof}
For $n = 1$, in which case $B U(1) \simeq \mathbb{C}P^\infty$ is the infinite [[complex projective space]], we have (\href{complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace}{prop})

\begin{displaymath}
H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ]
  \,,
\end{displaymath}
where $c_1$ is the [[first Chern class]]. From here we proceed by [[induction]]. So assume that the statement has been shown for $n-1$.

Observe that the canonical map $B U(n-1) \to B U(n)$ has as [[homotopy fiber]] the [[n-sphere|(2n-1)sphere]] (\href{classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces}{prop.}) hence there is a [[homotopy fiber sequence]] of the form

\begin{displaymath}
S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n)
  \,.
\end{displaymath}
Consider the induced [[Thom-Gysin sequence]].

In odd degrees $2k+1 \lt 2n$ it gives the [[exact sequence]]

\begin{displaymath}
\cdots
   \to
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}}
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to 
  \cdots
  \,,
\end{displaymath}
where the right term vanishes by induction assumption, and the middle term since [[ordinary cohomology]] vanishes in negative degrees. Hence

\begin{displaymath}
H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n
\end{displaymath}
Then for $2k+1 \gt 2n$ the Thom-Gysin sequence gives

\begin{displaymath}
\cdots
   \to
  H^{2k+1-2n}(B U(n))
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to 
  \cdots
  \,,
\end{displaymath}
where again the right term vanishes by the induction assumption. Hence [[exact sequence|exactness]] now gives that

\begin{displaymath}
H^{2k+1-2n}(B U(n))
   \overset{}{\longrightarrow}
  H^{2k+1}(B U(n))
\end{displaymath}
is an [[epimorphism]], and so with the previous statement it follows that

\begin{displaymath}
H^{2k+1}(B U(n)) \simeq 0
\end{displaymath}
for all $k$.

Next consider the Thom Gysin sequence in degrees $2k$

\begin{displaymath}
\cdots
   \to
  \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}}
   \longrightarrow
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}}
  \to 
  \cdots
  \,.
\end{displaymath}
Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a [[short exact sequence]]

\begin{displaymath}
0   
   \to
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
    \to
  0
\end{displaymath}
for all $k$. In degrees $\bullet\leq 2n$ this says

\begin{displaymath}
0   
   \to
  \mathbb{Z}
   \overset{c_n \cup (-)}{\longrightarrow}
  H^{\bullet \leq 2n}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n}
    \to
  0
\end{displaymath}
for some [[Thom class]] $c_n \in H^{2n}(B U(n))$, which we identify with the next Chern class.

Since [[free abelian groups]] are [[projective objects]] in [[Ab]], their [[extensions]] are all split (the [[Ext]]-group out of them vanishes), hence the above gives a [[direct sum]] decomposition

\begin{displaymath}
\begin{aligned}
    H^{\bullet \leq 2n}(B U(n))
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle
    \\
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n}
  \end{aligned}
  \,.
\end{displaymath}
Now by another induction over these short exact sequences, the claim follows.

\end{proof}
\hypertarget{first_chern_class}{}\subsubsection*{{First Chern class}}\label{first_chern_class}

\begin{itemize}%
\item The [[first Chern class]] of a bundle $P$ is the class of its [[determinant line bundle]] $det P$

\begin{displaymath}
c_1(P) = [det P]
  \,.
\end{displaymath}
See [[determinant line bundle]] for more.



\end{itemize}
\hypertarget{SplittingPrinciple}{}\subsubsection*{{Splitting principle and Chern roots}}\label{SplittingPrinciple}

Under the [[splitting principle]] all Chern classes are determnined by [[first Chern classes]]:

Write $i \colon T \simeq U(1)^n \hookrightarrow U(n)$ for the [[maximal torus]] inside the [[unitary group]], which is the [[subgroup]] of [[diagonal matrix|diagonal]] unitary matrices. Then

\begin{displaymath}
H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})
\end{displaymath}
is the [[polynomial ring]] in $n$ [[generators]] (to be thought of as the universal [[first Chern classes]] $c_i$ of each copy of $B U(1)$; equivalently as the \href{group+character#RelationToChernRootsAndSplittingPrinciple}{weights} of the [[group characters]] of $U(n)$) which are traditionally written $x_i$:

\begin{displaymath}
H^\bullet(B U(1)^n, \mathbb{Z})  \simeq \mathbb{Z}[x_1, \cdots, x_n]
  \,.
\end{displaymath}
Write

\begin{displaymath}
B i \;\colon\; B U(1)^n \to B U(n)
\end{displaymath}
for the induced map of [[deloopings]]/[[classifying spaces]], then the $k$-universal Chern class $c_k \in H^{2k}(B U(n), \mathbb{Z})$ is uniquely characterized by the fact that its pullback to $B U(1)^n$ is the $k$th [[elementary symmetric polynomial]] $\sigma_k$ applied to these first Chern classes:

\begin{displaymath}
(B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n)
  \,.
\end{displaymath}
Equivalently, for $c = \sum_{i = 1}^n c_k$ the formal sum of all the Chern classes, and using the fact that the [[elementary symmetric polynomials]] $\sigma_k(x_1, \cdots, k_n)$ are the degree-$k$ piece in $(1+x_1) \cdots (1+x_n)$, this means that

\begin{displaymath}
(B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n)
  \,.
\end{displaymath}
Since here on the right the first Chern classes $x_i$ appear as the [[roots]] of the Chern polynomial, they are also called \textbf{Chern roots}.

See also at \emph{\href{splitting+principle#ComplexVectorBundleAndTheirChernRoots}{splitting principle -- Examples -- Complex vector bundles and their Chern roots}}.

(e.g. \hyperlink{Kochmann96}{Kochmann 96, theorem 2.3.2}, \hyperlink{tomDieck08}{tom Dieck 08, theorem 19.3.2})

\begin{lemma}
\label{FromBUnTOBU1nPullbackInCohomologyIsInjective}\hypertarget{FromBUnTOBU1nPullbackInCohomologyIsInjective}{}
For $n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]]

\begin{displaymath}
\left( 
    B \iota_n
  \right)
  \;\colon\;
   H^\bullet( B U(n); \mathbb{Z} )
   \longrightarrow
   H^\bullet( B U(1)^n; \mathbb{Z} )
\end{displaymath}
is a [[monomorphism]].

\end{lemma}
A \textbf{[[proof]]} of lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective} , via analysis of the [[Serre spectral sequence]] of $U(n)/U(1)^n \to B U(1)^n \to B U(n)$ is indicated in (\hyperlink{Kochmann96}{Kochmann 96, p. 40}). A proof via [[Becker-Gottlieb transfer|transfer]] of the [[Euler class]] of $U(n)/U(1)^n$, following (\hyperlink{Dupont78}{Dupont 78, (8.28)}), is indicated at \emph{[[splitting principle]]} (\href{splitting+principle#InjectivityOfPullbackInCohomologyToBT}{here}).

\begin{prop}
\label{SplittingPrincipleForChernClasses}\hypertarget{SplittingPrincipleForChernClasses}{}
For $k \leq n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]] is of the form

\begin{displaymath}
(B i_n)^\ast 
  \;\colon\;
  \mathbb{Z}[c_1, \cdots, c_k]
  \longrightarrow
  \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]
\end{displaymath}
and sends the $k$th Chern class $c_k$ (def. \ref{GeneratorsOfCohomologyOfBunChernClasses}) to the $k$th [[elementary symmetric polynomial]] in the $n$ copies of the [[first Chern class]]:

\begin{displaymath}
(B i_n)^\ast 
    \;\colon\;
  c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n )
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
First consider the case $n = 1$.

The [[classifying space]] $B U(1)$ is equivalently the infinite [[complex projective space]] $\mathbb{C}P^\infty$. Its [[ordinary cohomology]] is the [[polynomial ring]] on a single generator $c_1$, the [[first Chern class]] (\href{complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace}{prop.})

\begin{displaymath}
H^\bullet(B U(1))
    \simeq
  \mathbb{Z}[ c_1 ]
  \,.
\end{displaymath}
Moreover, $B i_1$ is the identity and the statement follows.

Now by the [[Künneth theorem]] for ordinary cohomology (\href{K%C3%BCnneth+theorem#KunnethInOrdinaryCohomology}{prop.}) the cohomology of the [[Cartesian product]] of $n$ copies of $B U(1)$ is the [[polynomial ring]] in $n$ generators

\begin{displaymath}
H^\bullet(B U(1)^n)
   \simeq
  \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n]
  \,.
\end{displaymath}
By prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} the domain of $(B i_n)^\ast$ is the [[polynomial ring]] in the Chern classes $\{c_i\}$, and by the previous statement the codomain is the polynomial ring on $n$ copies of the first Chern class

\begin{displaymath}
(B i_n)^\ast
    \;\colon\;
   \mathbb{Z}[ c_1, \cdots, c_n ]
     \longrightarrow
   \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ]
  \,.
\end{displaymath}
This allows to compute $(B i_n)^\ast(c_k)$ by [[induction]]:

Consider $n \geq 2$ and assume that $(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)})$. We need to show that then also $(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n)$.

Consider then the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    B U(1)^{n-1}
      &\overset{ B i_{n-1} }{\longrightarrow}&
    B U(n-1)
    \\
    {}^{\mathllap{B j_{\hat t}}}\downarrow
      &&
    \downarrow^{\mathrlap{B i_{\hat t}}}
    \\
    B U(1)^n
     &\underset{B i_n}{\longrightarrow}&
    B U(n)
  }
\end{displaymath}
where both vertical morphisms are induced from the inclusion

\begin{displaymath}
\mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n
\end{displaymath}
which omits the $t$th coordinate.

Since two embeddings $i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n)$ differ by [[conjugation]] with an element in $U(n)$, hence by an [[inner automorphism]], the maps $B i_{\hat t_1}$ and $B_{\hat i_{t_2}}$ are [[homotopy|homotopic]], and hence $(B i_{\hat t})^\ast = (B i_{\hat n})^\ast$, which is the morphism from prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}.

By that proposition, $(B i_{\hat t})^\ast$ is the identity on $c_{k \lt n}$ and hence by induction assumption

\begin{displaymath}
\begin{aligned}
    (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n}
    &=
    (B i_{n-1})^\ast c_{k \lt n}
    \\
    = 
    \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n )
  \end{aligned}
  \,.
\end{displaymath}
Since pullback along the left vertical morphism sends $(c_1)_t$ to zero and is the identity on the other generators, this shows that

\begin{displaymath}
(B i_n)^\ast(c_{k \lt n})
    \simeq
   \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n)
   \;\;
   mod (c_1)_t
  \,.
\end{displaymath}
This implies the claim for $k \lt n$.

For the case $k = n$ the commutativity of the diagram and the fact that the right map is zero on $c_n$ by prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} shows that the element $(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0$ for all $1 \leq t \leq n$. But by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective} the morphism $(B i_n)^\ast$, is injective, and hence $(B i_n)^\ast(c_n)$ is non-zero. Therefore for this to be annihilated by the morphisms that send $(c_1)_t$ to zero, for all $t$, the element must be proportional to all the $(c_1)_t$. By degree reasons this means that it has to be the product of all of them

\begin{displaymath}
\begin{aligned}
    (B i_n)^{\ast}(c_n) 
     & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n
     \\
     & = \sigma_n( (c_1)_1, \cdots, (c_1)_n )
  \end{aligned}
  \,.
\end{displaymath}
This completes the induction step.

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

\begin{prop}
\label{WhitneySumChernClasses}\hypertarget{WhitneySumChernClasses}{}
For $k\leq n \in \mathbb{N}$, consider the canonical map

\begin{displaymath}
\mu_{k,n-k}
    \;\colon\;
  B U(k) \times B U(n-k)
    \longrightarrow
  B U(n)
\end{displaymath}
(which classifies the [[Whitney sum]] of [[complex vector bundles]] of [[rank]] $k$ with those of rank $n-k$). Under pullback along this map the universal [[Chern classes]] (prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}) are given by

\begin{displaymath}
(\mu_{k,n-k})^\ast(c_t)
  \;=\;
  \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i}
  \,,
\end{displaymath}
where we take $c_0 = 1$ and $c_j = 0 \in H^\bullet(B U(r))$ if $j \gt r$.

So in particular

\begin{displaymath}
(\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k}
  \,.
\end{displaymath}
\end{prop}
e.g. (\hyperlink{Kochmann96}{Kochmann 96, corollary 2.3.4})

\begin{proof}
Consider the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    H^\bullet( B U(n) )
      &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}&
    H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) )
    \\
    {}^{\mathllap{\mu_k^\ast}}\downarrow
      &&
    \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }}
    \\
    H^\bullet( B U(1)^n )
     &\simeq&
    H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} )
  }
  \,.
\end{displaymath}
This says that for all $t$ then

\begin{displaymath}
\begin{aligned}
    (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
    & =
    \mu^\ast_n(c_t)
    \\
    & = \sigma_t((c_1)_1, \cdots, (c_1)_n)
  \end{aligned}
  \,,
\end{displaymath}
where the last equation is by prop. \ref{SplittingPrincipleForChernClasses}.

Now the [[elementary symmetric polynomial]] on the right decomposes as required by the left hand side of this equation as follows:

\begin{displaymath}
\sigma_t((c_1)_1, \cdots, (c_1)_n)
    \;=\; 
  \underoverset{r = 0}{t}{\sum}
  \sigma_r((c_1)_1, \cdots, (c_1)_{n-k})
  \cdot
  \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n )
  \,,
\end{displaymath}
where we agree with $\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0$ if $q \gt p$. It follows that

\begin{displaymath}
(\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
  = 
  (\mu_k^\ast \otimes \mu_{n-k}^\ast)
  \left(
    \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r}
  \right)
  \,.
\end{displaymath}
Since $(\mu_k^\ast \otimes \mu_{n-k}^\ast)$ is a monomorphism by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective}, this implies the claim.

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

\hypertarget{ChernClassesOfLinearRepresentations}{}\subsubsection*{{Chern classes of linear representations}}\label{ChernClassesOfLinearRepresentations}

Under the [[Atiyah-Segal completion]] map [[linear representations]] of a [[group]] $G$ induce K-theory classes on the [[classifying space]] $B G$. Their Chern classes are hence invariants of the [[linear representations]] themselves.

See at \emph{[[characteristic class of a linear representation]]} for more.

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

\begin{itemize}%
\item [[Chern root]]


\item [[Pontryagin class]]


\item [[Stiefel-Whitney class]]


\item [[Chern character]]



\end{itemize}
In [[Yang-Mills theory]] field configurations with non-vanishing [[second Chern class]] (and minimal energy) are called [[instantons]]. The second Chern class is the \emph{[[instanton number]]} . For more on this see at \emph{\href{BPTS-instanton#FromTheMathsToThePhysicsStory}{SU(2)-instantons from the correct maths to the traditional physics story}}.

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

Original articles include

\begin{itemize}%
\item [[A. Grothendieck]], \emph{La th\'e{}orie des classes de Chern}, Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de France \textbf{86} (1958), p. 137--154, \href{http://www.numdam.org/item?id=BSMF_1958__86__137_0}{numdam}

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], chapter IX of volume II of \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973)


\item [[John Milnor]], [[Jim Stasheff|James D. Stasheff]], \emph{Characteristic Classes}, Annals of Mathematics Studies 76, Princeton University Press (1974).


\item [[Stanley Kochmann]], section 2.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Tammo tom Dieck]], \emph{Algebraic topology}, EMS 2008



\end{itemize}
With an eye towards [[mathematical physics]]:

\begin{itemize}%
\item Gerd Rudolph, Matthias Schmidt, Def. 4.2.2 of \emph{Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields}, Theoretical and Mathematical Physics series, Springer 2017 (\href{https://link.springer.com/book/10.1007/978-94-024-0959-8}{doi:10.1007/978-94-024-0959-8})

\end{itemize}
A brief introduction is in chapter 23, section 7

\begin{itemize}%
\item [[Peter May]], \emph{A concise course in algebraic topology} (\href{http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf}{pdf})

\end{itemize}
For [[Conner-Floyd Chern classes]] in [[complex oriented cohomology theory]]:

\begin{itemize}%
\item [[Frank Adams]], part II.2 and part III.10 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, 2010, lecture 4 (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf}{pdf}) and lecture 5 (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Dupont, \emph{Curvature and characteristic classes}, Springer 1978

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

[[!redirects universal Chern class]] [[!redirects universal Chern classes]]

[[!redirects second Chern class]]

[[!redirects Chern root]] [[!redirects Chern roots]]

[[!redirects total Chern class]] [[!redirects total Chern classes]]



\end{document}