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\begin{document}

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\section*{Serre spectral sequence}

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

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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{InRelativeCohomology}{In relative cohomology}\dotfill \pageref*{InRelativeCohomology} \linebreak
\noindent\hyperlink{in_equivariant_cohomology}{In equivariant cohomology}\dotfill \pageref*{in_equivariant_cohomology} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{consequences}{Consequences}\dotfill \pageref*{consequences} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_equivariant_cohomology_2}{In equivariant cohomology}\dotfill \pageref*{in_equivariant_cohomology_2} \linebreak


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

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

The \emph{Serre spectral sequence} or \emph{Leray-Serre spectral sequence} is a [[spectral sequence]] for computation of [[ordinary cohomology]] ([[ordinary homology]]) of [[topological spaces]] in a [[Serre fibration|Serre]]-[[fiber sequence]] of [[topological spaces]].

Given a [[homotopy fiber sequence]]

\begin{displaymath}
\itexarray{
     F &\longrightarrow& E
     \\
     && \downarrow^{\mathrlap{p}}
     \\
     && X
  }
\end{displaymath}
over a simply connected space $X$, then the corresponding cohomology Serre spectral sequence looks like

\begin{displaymath}
E_2^{p,q}= H^p(X, H^q(F)) \Rightarrow H^{p+q}(E)
  \,.
\end{displaymath}
\hypertarget{in_generalized_cohomology}{}\subsubsection*{{In generalized cohomology}}\label{in_generalized_cohomology}

The generalization of this from [[ordinary cohomology]] to [[generalized (Eilenberg-Steenrod) cohomology]] is the \emph{[[Atiyah-Hirzebruch spectral sequence]]}, see there for details.

\hypertarget{InRelativeCohomology}{}\subsubsection*{{In relative cohomology}}\label{InRelativeCohomology}

There are two kinds of \textbf{relative Serre spectral sequences}.

For $F \to E \to X$ as above and $A \hookrightarrow X$ a subspace, the induced restriction of the fibration

\begin{displaymath}
\itexarray{   
    F & \simeq & F
    \\
    \downarrow && \downarrow
    \\
    p^{-1}(A) &\longrightarrow& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    A &\hookrightarrow& X
  }
\end{displaymath}
induces a spectral sequence in [[relative cohomology]] of the base space of the form

\begin{displaymath}
E_2^{p,q}
  =
  H^p(X,A; H^q(F))
    \;\Rightarrow\;
  H^\bullet(E, p^{-1}(A))
  \,.
\end{displaymath}
(e.g. \hyperlink{Davis91}{Davis 91, theorem 9.33})

Conversely, for

\begin{displaymath}
\itexarray{   
    F' & \hookrightarrow & F
    \\
    \downarrow && \downarrow
    \\
    E' &\hookrightarrow& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    X &\hookrightarrow& X
  }
\end{displaymath}
a sub-fibration over the same base, then this induces a spectral sequence for [[relative cohomology]] of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:

\begin{displaymath}
E^{p,q}_2 = H^p(X; H^q(F,F'))
    \;\Rightarrow\;
  H^\bullet(E,E')
  \,.
\end{displaymath}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, theorem 2.6.3}, \hyperlink{Davis91}{Davis 91, theorem 9.34})

\hypertarget{in_equivariant_cohomology}{}\subsubsection*{{In equivariant cohomology}}\label{in_equivariant_cohomology}

There is also a generalization to [[equivariant cohomology]]: for \href{Mackey+functor#Cohomology}{cohomology with coefficients in a Mackey functor} with[[RO(G)-grading]] for [[representation spheres]] $S^V$, then for $E \to X$ an $F$-fibration of [[topological G-spaces]] and for $A$ any $G$-[[Mackey functor]], the equivariant Serre spectral sequence looks like (\hyperlink{Kronholm10}{Kronholm 10, theorem 3.1}):

\begin{displaymath}
E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(E,A)
  \,,
\end{displaymath}
where on the left in the $E_2$-page we have [[ordinary cohomology]] with [[coefficients]] in the genuine equivariant cohomology groups of the fiber.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

For details on the plain Serre spectral sequence see at \emph{[[Atiyah-Hirzebruch spectral sequence]]} and take $E = H R$ to be ordinary cohomology.

\hypertarget{consequences}{}\subsection*{{Consequences}}\label{consequences}

\begin{itemize}%
\item [[Serre long exact sequence]]


\item [[Gysin sequence]]



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

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

The original article is

\begin{itemize}%
\item [[Jean-Pierre Serre]], \emph{Homologie singuli\'e{}re des espaces fibr\'e{}s} Applications, Ann. of Math. 54 (1951),

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Alan Hatcher]], \emph{\href{http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html}{Spectral sequences in algebraic topology}} \emph{I: The Serre spectral sequence} (\href{http://www.math.cornell.edu/~hatcher/SSAT/SSch1.pdf}{pdf})


\item [[Stanley Kochmann]], section 2.2. of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item Davis, \emph{Lecture notes in algebraic topology}, 1991



\end{itemize}
Lecture notes etc. includes

\begin{itemize}%
\item [[Greg Friedman]], \emph{Some extremely brief notes on the Leray spectral sequence} (\href{http://faculty.tcu.edu/richardson/Seminars/Gregspecseq.pdf}{pdf})

\end{itemize}
Discussion in [[homotopy type theory]] includes

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{http://homotopytypetheory.org/2013/08/08/spectral-sequences/}{Spectral Sequences}}, 2013

\end{itemize}
and implementation in [[Lean]] is in

\begin{itemize}%
\item [[Floris van Doorn]], [[Egbert Rijke]], [[Ulrik Buchholtz]], [[Favonia]], [[Steve Awodey]], [[Jeremy Avigad]], [[Mike Shulman]], [[Jonas Frey]], \emph{Spectral} (\href{https://github.com/cmu-phil/Spectral}{github.com/cmu-phil/Spectral})

\end{itemize}
\hypertarget{in_equivariant_cohomology_2}{}\subsubsection*{{In equivariant cohomology}}\label{in_equivariant_cohomology_2}

In [[equivariant cohomology]], for [[Bredon cohomology]]:

\begin{itemize}%
\item [[Ieke Moerdijk]], J.-A. Svensson, \emph{The Equivariant Serre Spectral Sequence}, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 (\href{http://www.jstor.org/stable/2160037}{JSTOR})

\end{itemize}
and for genuine equivariant cohomology, i.e. for [[RO(G)-grading|RO(G)]]-graded \href{Mackey+functor#Cohomology}{cohomology with coefficients in a Mackey functor}:

\begin{itemize}%
\item [[William Kronholm]], \emph{The $RO(G)$-graded Serre spectral sequence}, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (\href{http://www.swarthmore.edu/NatSci/wkronho1/serre.pdf}{pdf}, \href{https://projecteuclid.org/euclid.hha/1296223823}{Euclid})

\end{itemize}
See also

\begin{itemize}%
\item [[Megan Shulman]], \emph{Equivariant Spectral Sequences for Local Coefficients} (\href{http://arxiv.org/abs/1005.0379}{arXiv:1005.0379})

\end{itemize}
[[!redirects Serre spectral sequences]]

[[!redirects Leray-Serre spectral sequence]] [[!redirects Leray-Serre spectral sequences]]



\end{document}