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\section*{splitting principle}

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

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

[[!include cohomology - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

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

\noindent\hyperlink{SplittingPrinciple}{Idea}\dotfill \pageref*{SplittingPrinciple} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{ComplexVectorBundleAndTheirChernRoots}{Complex vectors bundles and their Chern roots}\dotfill \pageref*{ComplexVectorBundleAndTheirChernRoots} \linebreak
\noindent\hyperlink{linear_representations_and_brauer_induction}{Linear representations and Brauer induction}\dotfill \pageref*{linear_representations_and_brauer_induction} \linebreak
\noindent\hyperlink{RealVectorBundles}{Real vector bundles}\dotfill \pageref*{RealVectorBundles} \linebreak
\noindent\hyperlink{genera_and_hirzebruch_characteristic_series}{Genera and Hirzebruch characteristic series}\dotfill \pageref*{genera_and_hirzebruch_characteristic_series} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[algebraic topology]], a \emph{splitting principle} for a [[classifying space]] $B G$ for some [[topological group]] $G$ ([[delooping]] of some [[∞-group]] $G$) is a map from a [[torus]]-classifying space

\begin{displaymath}
B (T^n) \overset{f}{\longrightarrow} B G
\end{displaymath}
such that the induced pullback of [[cohomology rings]] is an [[injective function]]

\begin{displaymath}
f^\ast \;\colon\; H^\bullet(B G) \hookrightarrow H^\bullet(B T^n)
     \,,
\end{displaymath}
hence allowing to view the cohomology of $G$-[[principal ∞-bundles]] in terms of that of plain [[torus]]-[[principal bundles]].

In particular for $G$ a classical [[compact Lie group]] such as the [[unitary group]], the splitting principle holds and allows to express [[Chern classes]] of [[complex vector bundles]] as algebraic expressions in just [[first Chern classes]] of [[complex line bundles]].

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

Let $G$ be a [[connected topological space|connected]] [[compact Lie group]] and write $U(1)^n \simeq T \stackrel{i}{\hookrightarrow} G$ for a [[maximal torus]].

Let $X$ be a [[connected topological space]] and $P \to X$ a $G$-[[principal bundle]] over $X$ [[classifying space|classified]] by a map $g \colon X \to B G$.

Then consider the [[coset space]] $G/T$ and the $G/T$-[[fiber bundle]] $Y\to X$ [[associated bundle|associated]] to $P$, this is equivalently the [[homotopy pullback]] in the diagram

\begin{displaymath}
\itexarray{
    Y &\stackrel{(g_1,\cdots, g_n)}{\longrightarrow}& B T & \simeq B U(1)^n 
    \\
    \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{B i}}
    \\
    X &\stackrel{g}{\longrightarrow}& B G
  }
  \,.
\end{displaymath}
This diagram shows that the pullback of the $G$-principal bundle $P \to X$ along $p$ to $Y$ is equivalently a $T$-principal bundle splitting as [[circle group]]-principal bundles classified by $(g_1, \cdots, g_n)$.

That this is a useful splitting is the content of:

\begin{theorem}
\label{GeneralizedSplittingPrinciple}\hypertarget{GeneralizedSplittingPrinciple}{}
\textbf{(generalized splitting principle)}

Let $R$ be a [[commutative ring]] in which a [[prime number]] $p$ is a [[unit]] if $H_\bullet(B G,\mathbb{Z})$ has a $p$-[[torsion subgroup]].

Then

\begin{enumerate}%
\item The $H^\bullet(B G,R)$-module $H^\bullet(B T, R)$ (via $B i^\ast$) is [[free module|free]] on the cohomology of $G/T$:

\begin{displaymath}
H^\bullet(B T,R)\simeq H^\bullet(B G,R) \otimes H^\bullet(G/T, R)
  \,;
\end{displaymath}

\item Analogously there is an isomorphism

\begin{displaymath}
H^\bullet(Y,R) \simeq H^\bullet(X,R)\otimes H^\bullet(G/T,R)
\end{displaymath}
and hence $p^\ast$ is the canonical inclusion (and hence in particular is an [[injection]])

\begin{displaymath}
p^\ast \;\colon\;  H^\bullet(X,R)\hookrightarrow H^\bullet(Y,R)
  \,.
\end{displaymath}


\end{enumerate}
\end{theorem}
In this general form this is due to (\hyperlink{May}{May}).

\begin{theorem}
\label{}\hypertarget{}{}
Since the elements

\begin{displaymath}
c \in H^\bullet(B G,R)
\end{displaymath}
are the [[universal characteristic classes]] of $G$-principal bundles with [[coefficients]] in $R$ (hence by [[Chern-Weil theory]] the [[invariant polynomials]] of the [[Lie algebra]] $\mathfrak{g}$ if $R$ has [[characteristic]]-0), theorem \ref{GeneralizedSplittingPrinciple} gives the following way to express the characteristic classes of $G$-principal bundles on $X$ by [[tuples]]

\begin{displaymath}
(c_1^1, \cdots, c_1^n) \coloneqq (B i)^\ast c
\end{displaymath}
of characteristic classes -- hence [[first Chern class|first Chern classes]] -- of just circle bundles ([[line bundles]]):

\begin{displaymath}
p^\ast (c(P)) \simeq (g_1^\ast c_1^1, \cdots, g_n^\ast c_1^n)
  \,.
\end{displaymath}
(Since $p^\ast$ is injective, this is a genuine characterization of $c(P)$).

\end{theorem}
\begin{remark}
\label{InjectivityOfPullbackInCohomologyToBT}\hypertarget{InjectivityOfPullbackInCohomologyToBT}{}
One way to see that $B i^*\colon H^*(B G) \to H^*(B T)$ is injective is by using [[Chern-Weil theory]] to recognise that this map is just $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$ for $G$ a compact [[Lie group]]. This tells us firstly that these cohomology rings are particularly nice.

One can define a [[Becker-Gottlieb transfer|transfer map]] $\tau\colon H^*(B T) \to H^*(B G)$ as in \href{http://mathoverflow.net/questions/61784/cohomology-of-bg-g-compact-lie-group/61943#61943}{this MO answer}, and then show, following (\hyperlink{Dupont78}{Dupont 1978}, chapter 8), that $\tau\circ B i^*\colon H^*(B G)\to H^*(B G)$ is multiplication by the [[Euler class]] $\chi(G/T)$. Thus if $\chi(G/T) \gt 0$ then $\tau\circ B i^*$ hence $B i^*$ is injective. One can calculate $\chi(G/T) = | N(T)/T |$, where $N(T)/T =: W_T$ is the [[Weyl group]] of the maximal torus $T$, using a [[Lefschetz trace formula|Lefshetz fixed point]]-argument, giving the result.

\end{remark}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{ComplexVectorBundleAndTheirChernRoots}{}\subsubsection*{{Complex vectors bundles and their Chern roots}}\label{ComplexVectorBundleAndTheirChernRoots}

For $G = U(n)$ the [[unitary group]], the [[universal characteristic classes]] are the [[Chern classes]] $c_k \in H^\bullet(B U(n), \mathbb{Z})$. By the discussion at \emph{\href{Chern+class#SplittingPrinciple}{Chern class -- Properties -- Splitting principle and Chern roots}. the universal splitting principle here says that}

\begin{displaymath}
(B i)^\ast(\sum_k c_k) = (1 + x_1) \cdots (1+ x_n)
  \,,
\end{displaymath}
where the $x_i \in H^\bullet(B U(1)^n , \mathbb{Z})$ are the universal characteristic classes of the [[maximal torus]], hence are $n$ incarnations of the universal [[first Chern class]] (equivalently: the \href{group+character#RelationToChernRootsAndSplittingPrinciple}{weights} of the [[group characters]] of $U(n)$). It follows that every [[complex vector bundle]] $\xi$ of [[rank]] $n$ over a space $X$ when pulled back to its [[flag space]] bundle decomposes as a [[direct sum]] of [[complex line bundles]] $\zeta_i$ and has [[Chern classes]] $c_k$ expressed in terms of the [[first Chern classes]] of these line bundles as

\begin{displaymath}
c_k(p^\ast \xi ) = \sigma_k(c_1(\zeta_1), \cdots, c_n(\zeta_n))
  \,.
\end{displaymath}
This case is what is traditionally is often meant by default by the ``splitting principle''.

For the [[special unitary group]] the situation is the same, only that here the splitting is into a sum of line bundles used [[tensor product]] is constrained to be trivializable.

\hypertarget{linear_representations_and_brauer_induction}{}\subsubsection*{{Linear representations and Brauer induction}}\label{linear_representations_and_brauer_induction}

The [[Brauer induction theorem]] may be regarded as the splitting principle for [[linear representations]] (\hyperlink{Symonds91}{Symonds 91}), see also at \emph{[[characteristic classes of linear representations]]},

\hypertarget{RealVectorBundles}{}\subsubsection*{{Real vector bundles}}\label{RealVectorBundles}

Under the \emph{\href{Pontryagin+class#FurtherRelationToChernClasses}{Relation between Pontryagin classes and Chern classes}} the above translates into a splitting principle also for [[real vector bundles]].

\hypertarget{genera_and_hirzebruch_characteristic_series}{}\subsubsection*{{Genera and Hirzebruch characteristic series}}\label{genera_and_hirzebruch_characteristic_series}

The basic theorem of [[Hirzebruch series]] expresses [[genera]] via the splitting principle. The Hirzebruch characteristic series $K_\phi$ is a [[series]] in a single variable $x = c_1(L)$, to be thought of as the [[first Chern class]] of the universal [[complex line bundle]] over $B U(1)$.

The Hirzebruch formula for the value of the [[genus]] $\phi$ on an [[orientation|oriented]] [[manifold]] $X$

\begin{displaymath}
\phi(X) = \langle K_\phi(T X), [X]\rangle
\end{displaymath}
denotes the pairing of that class of the [[tangent bundle]] with the [[fundamental class]] which under the splitting principle pulls back on the [[flag space]] bundle to the class $\prod_k K_\phi(x_k)$ of the corresponding direct sum of line bundles.

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

\begin{itemize}%
\item [[Snaith's theorem]]

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

\begin{itemize}%
\item [[Peter May]], \emph{A note on the splitting principle}, Topology and its Applications Volume 153, Issue 4, 1 November 2005, Pages 605-609 (\href{http://www.math.uchicago.edu/~may/PAPERS/Split.pdf}{pdf}, \href{https://doi.org/10.1016/j.topol.2005.02.007}{doi:10.1016/j.topol.2005.02.007})


\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 10.2 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})



\end{itemize}
Discussion in the derivation of [[Chern classes]] and [[Stiefel-Whitney classes]] includes

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

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Splitting_principle}{Splitting principle}}


\item \emph{Notes on the splitting principle} (\href{http://www.math.sunysb.edu/~azinger/mat566/splitting.pdf}{pdf})


\item Johan L. Dupont, \emph{Curvature and Characteristic Classes}, Lecture Notes in Mathematics, \textbf{640} (1978) doi:\href{http://dx.doi.org/10.1007/BFb0065364}{10.1007/BFb0065364}.



\end{itemize}
Discussion in the context of [[complex oriented cohomology theory]] and their [[generalized Chern classes]] includes

\begin{itemize}%
\item \hyperlink{Kochmann96}{Kochmann 96, section 4.3}


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



\end{itemize}
More expository discussion in the context of [[characteristic classes]] with applications in [[mathematical physics]] is in

\begin{itemize}%
\item [[Yang Zhang]], \emph{A brief introduction to characteristic classes from the differentiable viewpoint}, 2011 (pdfrief introduction to characteristic classes from the differentiable viewpoint.pdf))

\end{itemize}
The generalization to a splitting principle for [[twisted vector bundles]] ([[twisted cohomology]]) is discussed (in terms of [[bundle gerbe modules]]) in

\begin{itemize}%
\item [[Atsushi Tomoda]], \emph{On the splitting principle of bundle gerbe modules}, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (\href{https://projecteuclid.org/euclid.ojm/1174324334}{Euclid}, talk slides \href{http://ton.prosou.nu/official/ryousi2005.pdf}{pdf})

\end{itemize}
For [[characteristic classes of linear representations]]:

\begin{itemize}%
\item [[Peter Symonds]], \emph{A splitting principle for group representations}, Comment. Math. Helv. (1991) 66: 169 (\href{https://doi.org/10.1007/BF02566643}{doi:10.1007/BF02566643})

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



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