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\begin{document}

\title[SSS]{Serre Spectral Sequences and Applications}
%\subjclass[2000]{Primary 55T10; Secondary 55-01}
%keywords{Serre Spectral Sequences}
\author{David Brown}
\address{Department of Mathematics, University of California, Berkeley, CA  
94720-3840, USA}
\email{brownda@math.berkeley.edu}

\date{May 15, 2006}
 
\begin{abstract}
     Spectral sequences are a key theoretical and computational tool in
     algebraic topology. This paper is a tutorial essay,
     intended for students and researchers as a quick
     introduction, providing a few nice examples, applications, and a 
     short list of references for further inquiry.
\end{abstract}

\maketitle
\tableofcontents
%****************************************************************************
\section{Introduction}
\label{introduction}

 Algebraic topology focuses on the properties
 of spaces and maps between spaces which are invariant under
 `wiggling'. Some central tools are algebraic gadgets such as homotopy,
 homology, and cohomology; these give a way to distinguish spaces and 
 occasionally completely characterize spaces. Serre Spectral Sequences
 are a powerful theoretical and computational tool with numerous
 applications to Algebraic Topology. Indeed, a main application is the
 computation of various homotopy groups of spheres, and we will
 demonstrate this by the calculation of $\pi_{4}(S^{3})$.
 
 The exposition will first introduce the notion of a spectral sequence.
 As an inspiring example we will calculate 
 $\pi_{4}(S^{3})$. We next shift gears and explain an
 application of spectral sequences to the author's first love, number theory.
 Finally, we conclude with a few pages of abstract nonsense~(see \cite[Part
 Four, Introduction]{Lang} for a precise definition of abstract
 nonsense),
 illuminating the notion of a spectral sequence as we get our hands dirty 
 with the pure homological algebra at its core and discover its
 essence.
 
 We use as our main reference for 
 any definitions or theorems Allen Hatcher's \emph{Algebraic Topology} \cite{HAT}. The more
 topological pieces of this exposition follows Hatcher's
 upcoming book \emph{Serre Spectral Sequences in Algebraic Topology}  \cite{SSS}.
 The more abstract parts follow
 McCleary's book \emph{A User's Guide to Spectral Sequences} \cite{user}, with insights from 
 Chow's article \emph{You Could Have Invented Spectral Sequences} \cite{Chow}.\\
 
 \remark Basic definitions from algebraic topology
 will be occasionally given and occasionally omitted, the
 idea being that certain notions are really prerequisite for this
 exposition to have any meaning. For example, it would be silly to
 define notions such as Fibration or Cohomology because if you don't 
 already know and care about these notions then briefly defining them 
 and proving enough properties to make this exposition precise would
 be pointless, because you won't learn anything or care about Serre Spectral 
 Sequences.
 Conversely, it is easy to forget precise statements of theorems like 
 Universal Coefficients, so where appropriate I will state them, and
 in any case I will always try to give concise citations.

 %****************************************************************************
\section{What is a Spectral Sequence?}
The notion of a spectral sequence generalizes many typical algebraic
situations that pop up in topology. For instance, instead of a
pair $A \subset X$, consider a filtration
\begin{equation}
    X_{0} \subset X_{1} \subset X_{2} \subset \ldots \subset X_{i}
\subset \ldots \subset X. \label{fil}
\end{equation}
Then instead of the long exact sequence of the pair $(X,A)$, which relates
the cohomology 
rings $H^{i}(X)$, $H^{i}(A)$, and $H^{i}(X,A)$, we get an 
algebraic relation between the cohomology of everything appearing in
$(\ref{fil})$. Similarly one can generalize the Mayer-Vietoris
sequence of a decomposition $X = A \cup B$ to a covering by any
number of sets.
 
 
So again, what is a spectral sequence? Poetically, we begin with some
initial data (like a filtration of a topological space $X$)
and construct an infinite book. We can give each page the structure
of a graded complex. Then the $(p+1)^{\text{th}}$ page is formed from 
the homology of the $p^{\text{th}}$ page. Under suitable conditions
each entry of a page eventually stabilizes, and we can pass to a
limit, forming page $\infty$. And again under suitable conditions,
the entries on page $\infty$ tells us a lot about our initial
data (for instance, information about the cohomology ring $H^{*}(X)$). 

%****************************************************************************
\section{Theorems from Algebraic Topology} 
 Here we recall without proof standard theorems from algebraic
 topology that will be used later in this exposition.
 
%-----------------------------------------------------------------------------
 \begin{theorem} \label{ucoco} (Universal Coefficients for Cohomology)
     Let $C$ be a chain complex of free abelian groups with homology
     groups $H_{n}(C)$, and let $G$ be a group. Then the cohomology 
     groups of the chain complex $\text{Hom}(C_{n},G)$ fit into the
     split exact sequences
     \[
     0 \to \text{Ext}(H_{n-1}(C),G) \to H^{n}(C;G) \to
     \text{Hom}(H_{n}(C),G) \to 0.
     \]
     \end{theorem}
 In practice, the Ext term either vanishes or is easily computable.
 See \cite[Chapter 3, Section 
 1, Theorem 3.2]{HAT} for proofs and to further understand
 the Ext term.\\
 
 \begin{theorem}\label{Hurewicz} (Hurewicz)
     Let $X$ be a simply connected space, and let $i > 1$ be the first
     integer such that $\pi_{i}(X) \neq 0$. Then we have an
     isomorphism
     \[
     \pi_{i}(X) \cong H_{i}(X)
     \]
     and a surjection 
     \[
     \pi_{i+1}(X) \to H_{i+1}(X) \to 0.
     \]
     \end{theorem}
 See \cite[Chapter 4, Section 2, Theorem 4.32] {HAT} for a proof.\\

 \begin{theorem}\label{LES} (Long Exact Sequence of a
     Fibration)
 Let $F \to E \to B$ be a fibration (see
 \cite[Chapter 4, Section 3]{HAT} for a definition). Then there is a long exact sequence
 \[\cdots \to \pi_{n}(F) \to \pi_{n}(E) \to \pi_{n}(B) \to
 \pi_{n-1}(F) \to \cdots \to \pi_{0}(E) \to 0 .
 \]
     \end{theorem}
 See \cite[Chapter 4, Section 2, Theorem 4.41]{HAT} for a proof. 
     
     
%****************************************************************************
 \section{Main Application to Algebraic Topology: Cohomology Via
 Fibrations }\label{fibration}
 We begin with a (somewhat) concrete example. Consider a fibration 
 \[
 F \to X \to B.
 \]
 We will start by defining `page 2'; page 0 and 1 are interesting and
 will later be illuminating, but the journey from page 0 to 2 is best saved for the
 abstract nonsense of the appendix. Our goal will be to deduce 
 *something* about $H^{*}(X)$ based on information about $H^{*}(B)$ and
 $H^{*}(F)$.\\
 
 Set 
 \[
 E_{2}^{p,q} := \left\{\begin{array}{ll}
 H^{p}(B,H^{q}(F;G)) & \text{ $p,q \geq 0$ } \\
 0 & \text{ otherwise}  
 \end{array}\right.
 \]
 In the appendix we explain how to construct maps 
 $E_{2}^{p,q} \xrightarrow{\partial} E_{2}^{p+2,q-1}$
 giving $E_{2}^{\bullet}$ the structure of a graded complex:
 
 \[ 
 \xymatrix{
 \vdots    & \vdots   & \vdots & \vdots   & \vdots & \\
 E_{2}^{0,3}\ar[rrd] & E_{2}^{1,3} \ar[rrd]& E_{2}^{2,3}\ar[rrd] & E_{2}^{3,3} & E_{2}^{4,3} & 
 \cdots \\
 E_{2}^{0,2}\ar[rrd] & E_{2}^{1,2}\ar[rrd] & E_{2}^{2,2}\ar[rrd] & E_{2}^{3,2} & E_{2}^{4,2} & 
 \cdots \\
  E_{2}^{0,1}\ar[rrd] & E_{2}^{1,1}\ar[rrd] & E_{2}^{2,1} \ar[rrd]& E_{2}^{3,1} & E_{2}^{4,1} & 
 \cdots \\
 E_{2}^{0,0} & E_{2}^{1,0} & E_{2}^{2,0} & E_{2}^{3,0} & E_{2}^{4,0} & 
 \cdots \\  }
 \]\null
 
 
 We now take homology, setting
 \[
 E_{3}^{p,q} := \frac{\ker \left(E_{2}^{p,q} \xrightarrow{\partial}
 E_{2}^{p+2,q-1}\right)}{\im \left(E_{2}^{p-2,q+1} \xrightarrow{\partial}
 E_{2}^{p,q}\right)}
 \]
 to get page three. Similarly, we get maps 
 $E_{3}^{p,q} \xrightarrow{\partial} E_{3}^{p+3,q-2}$
 and a graded complex:
 \[ 
 \xymatrix{
 \vdots    & \vdots   & \vdots & \vdots   & \vdots & \\
 E_{3}^{0,3}\ar[rrrdd] & E_{3}^{1,3} \ar[rrrdd]& E_{3}^{2,3} & E_{3}^{3,3} & E_{3}^{4,3} & 
 \cdots \\
 E_{3}^{0,2}\ar[rrrdd] & E_{3}^{1,2}\ar[rrrdd] & E_{3}^{2,2} & E_{3}^{3,2} & E_{3}^{4,2} & 
 \cdots \\
  E_{3}^{0,1}\ar[rrrdd] & E_{3}^{1,1}\ar[rrrdd] & E_{3}^{2,1} & E_{3}^{3,1} & E_{3}^{4,1} & 
 \cdots \\
 E_{3}^{0,0} & E_{3}^{1,0} & E_{3}^{2,0} & E_{3}^{3,0} & E_{3}^{4,0} & 
 \cdots \\ 
 0 & 0 &  0 & 0 &  0 & \cdots
  }
 \]\null
 
 We take homology to get page 4. \\
 
 More generally we get maps 
 $E_{r}^{p,q} \xrightarrow{\partial} E_{r}^{p+r,q-r+1}$
 on the $r^{\text{th}}$ page and take homology to get the
 $(r+1)^{\text{th}}$ page.\\
 
 Observe that each $\partial$ eventually becomes trivial; for a fixed 
 $(p,q)$ and large $r$ the map 
 $E_{r}^{p,q} \xrightarrow{\partial} E_{r}^{p+r,q-r+1}$
 will have trivial co-domain, and the map 
 $E_{r}^{p-r,q+r-1} \xrightarrow{\partial} E_{r}^{p,q}$
 with have trivial domain. Thus for a fixed $(p,q)$ and large $r$
 (depending on $p$ and $q$) we have 
 \[
 E_{r}^{p,q} = E_{r+1}^{p,q} = E_{r+2}^{p,q} = \ldots 
 \]
 and so for such an $r$ we set $E^{\infty}_{p,q} := E_{r}^{p,q}$.
 
 The punchline is that with sufficient assumptions on our fibration, 
 the groups $E_{\infty}^{p,q}$, which we
 constructed from the data $H^{p}(B,H^{q}(F))$, are related to the
 groups $H^{p+q}(X)$. \\
 
 More precisely, let us specialize to the case of a fibration of
 CW-complexes $
 F \to X \xrightarrow{\pi} B$
 such that $B$ is simply-connected. The $p$ skeletons 
 \[
 B^{0} \subset B^{1} \subset \ldots \subset B^{p} \subset \ldots
 \subset B
 \]
 induce a filtration on $X$;  set $X^{p} := \pi^{-1}(B^{p})$. Then
 \[
 X^{0} \subset X^{1} \subset \ldots \subset X^{p} \subset \ldots
 \subset X
 \]
 is a filtration. The inclusions $\xymatrix{X_{p} \ar@{^{(}->}[r]&X}$ 
 induce maps $H^{n}(X) \to H^{n}(X_{p})$ and so we set 
 \[
 F_{n}^{p}:= \ker H^{n}(X) \to H^{n}(X_{p}).
 \]
 This gives a filtration of groups
 \[
 0 \subset F^{n}_{n}\subset \ldots \subset
 F^{n}_{p}
 \subset \ldots \subset F^{n}_{0} = H^{n}(X).
 \]
 Then the theorem of Serre says
 \[
 E_{\infty}^{p,n-p} \cong F^{n}_{p}/F^{n}_{p+1}.
 \]
 See \cite{SSS} for a more lengthy discussion and more general
 theorems.
%****************************************************************************
 \section{Additional Data} 
 \label{data}
 
 \subsection{\textbf{Naturality}}\label{Naturality} Given maps of fibrations, i.e. commutative
 diagrams 
 \[
 \xymatrix{
 F \ar[r]\ar[d] &X \ar[r]\ar[d] & B \ar[d]^{\phi}\\
 F' \ar[r] &X' \ar[r] & B' 
 }
 \]
 such that the fiber over $b \in B$ maps to the fiber over $\phi(b)
 \in B'$, we get maps
 \[
 E_{r}^{p,q} \to E_{r}^{'\, p,q}
 \]
 (including the case $r = \infty$) which respect all differentials and
 isomorphisms.\\
 
 \subsection{\textbf{Derivations}}\label{Derivations} Often we have extra structure, which the
 spectral sequence respects. For example, when $X$
 is a topological space $H^{*}(X)$ has a ring structure induced
 by the cup product, and the differentials $\partial$ respect
 this ring structure. More precisely, $\partial$ is a
 derivation, and the Leibintz rule is satisfied:
 \[
 \partial(\alpha \cdot \beta) = 
 \partial(\alpha) \cdot \beta
  + (-1)^{p+q} \alpha \partial (\beta).
 \]
 More precisely, there are bilinear products $E_{r}^{p,q} \times
 E_{r}^{s,t} \to E_{r}^{p+s,q+t}$ which are just the standard cup
 product when $r = 2$. As we will soon see, this extra structure can allow us to compute information about 
 the spectral sequence that we otherwise could not. See
 \cite[Chapter 1, Section 2]{SSS} for proofs and further discussion.\\
 
%----------------------------------------------------------------------------
 \section{Eilenberg-MacLane Spaces}
 Given a group $G$ and an integer $n$, one can use CW-complexes to elegantly construct
 spaces $K(G,n)$ with the nice property that
 \[
 \pi_{i}\left(K(G,n)\right) = \left\{\begin{array}{ll}
G & \text{if $i = 0,n$ } \\
0 & \text{ otherwise}  \end{array}\right. .
 \]
 The construction is simple; choose a presentation for $G$ (generators
and relations), and create a CW-complex with one zero-cell, one n-cell for each 
generator, one (n+1)-cell for each relation, then (n+2)-cells to kill 
of any homotopy's added by the (n+1)-cells, and so on.\\

This construction tends to have cells in every dimension, giving a
possibility of non-trivial homology and cohomology groups of every
dimension. Indeed, they are often non-trivial! While we
know from the Hurewicz theorem (\ref{Hurewicz}) and Universal
Coefficients for cohomology (\ref{ucoco}) that \label{bottom}
\[
G = \pi_{n}(K(G,n)) \cong H_{n}(K(G,n)),
\]
\[
H^n(K(G,n)) \cong
\text{Hom}(H_{n}(K(G,n)),\Z) \cong \text{Hom}\left(G,\Z\right),
\]
this is pretty much all we know in general, and calculating higher
homotopy groups is one task we will now demonstrate using a spectral
sequence.\\

See \cite[4.2]{HAT} for more facts on Eilenberg-MacLane Spaces.


%----------------------------------------------------------------------------
%----------------------------------------------------------------------------
\subsection{$K(\Z,2)$}
Using the fact that $S^{1}$ is a $K(\Z,1)$, we calculate
$H^{*}(K(\Z,2))$.\\

Consider the pathspace fibration \label{pathspace} \cite[4.3]{HAT} $F \to P \to B$,
where $B$ is a $K(\Z,2)$. $P$ is contractible, so from the long exact 
sequence of a fibration (\ref{LES}) we get for $i > 0$ that 
\[
\pi_{i}(K(\Z,1)) \cong \pi_{i+1}(K(\Z,2))
\]
and conclude that $F$ is a $K(\Z,1)$:
 \[
 \pi_{i}\left(F\right) = \left\{\begin{array}{ll}
\Z & \text{if $i = 0,1$ } \\
0 & \text{ otherwise}  \end{array}\right. . 
 \]
 Setting 
 \[
E_{2}^{p,q} := H^{p}(K(\Z,2); H^{q}(S^{1})) = \left\{\begin{array}{ll}
H^{p}(S^{1},\Z) & \text{if $q = 0,1$ } \\
0 & \text{ otherwise}  \end{array}\right. ,  
\]
 we get the following for page one:
\[
\begin{tabular}{c|ccccc}
2    & 0  & 0  & 0  & 0  & 0   \\
1    & $H^{0}(K(\Z,2))$ & $H^{1}(K(\Z,2))$ &$H^{2}(K(\Z,2))$ &$H^{3}(K(\Z,2))$
&$H^{4}(K(\Z,2))$   \\
0    & $H^{0}(K(\Z,2))$ & $H^{1}(K(\Z,2))$ &$H^{2}(K(\Z,2))$ &$H^{3}(K(\Z,2))$
&$H^{4}(K(\Z,2))$  \\
    \hline
     & 0 & 1 & 2 & 3 & 4 \\
\end{tabular}.\]
The differentials on page 2 are 
\[
 H^{r}(K(\Z,2)) \to  H^{r+2}(K(\Z,2)).
\]
Furthermore, these are the only differentials that have a chance of
killing anything (because all other differentials travel at least two 
rows and thus start or end with a 0), and so page $3$ will be the same as page $\infty$.
Consider also that the middle space is contractible, so that each
$E_{\infty}^{p,q}$ must be 0. We finally conclude that each
differential is an isomorphism, and thus 
\[
 H^{r}(K(\Z,2)) \cong \left\{\begin{array}{ll}
\Z & \text{$r$ even } \\
0 & \text{ $r$ odd}  \end{array}\right. . 
\]

%----------------------------------------------------------------------------
%----------------------------------------------------------------------------
\subsection{$K(\Z,3)$}
Similarly, we have a pathspace fibration $F \to P \to B$,
where $B$ is a $K(\Z,3)$, and again $P$ is contractible and (from the 
long exact sequence of a fibration (\ref{LES})) $F$ is a $K(\Z,2)$
(and so homotopy equivalent to $\C P^{\infty}$). The Hurewicz theorem (\ref{Hurewicz}) 
 tells us that $H^{3}(K(\Z,3)) = \Z$. Furthermore, below (\ref{sphere})
 we will see that we can find a model of $K(\Z,3)$ with no 4-cells, so
 $H^{4}(K(\Z,3))$ and $H_{4}(K(\Z,3)) = 0$. From Universal
 Coefficients (\ref{ucoco}) we see that 
 \[
 H^{5}(K(\Z,3)) = \text{Ext}(H_{4}(K(\Z,3),\Z ) \oplus \text{Hom}
 (H_{5}(K(\Z,3)),\Z) = 0. 
 \]
 
 Setting $E_{2}^{p,q}:=H^{p}(K(\Z,3),H^{q}(\C P^{\infty}))$, we
 get the following for page two:
  \[
\begin{tabular}{c|ccccccc}
5    & 0    & 0  & 0  & 0       & 0 & 0 & 0  \\
4    & $\Z$ & 0  & 0  & $\Z$    & 0 & $H^{5}(K(\Z,3))$ & $H^{6}(K(\Z,3))$  \\
3    & 0    & 0  & 0  & 0       & 0 & 0 & 0 \\
2    & $\Z$ & 0  & 0  & $\Z$    & 0 & $H^{5}(K(\Z,3))$ & $H^{6}(K(\Z,3))$  \\
1    & 0    & 0  & 0  & 0       & 0 & 0 & 0  \\
0    & $\Z$ & 0  & 0  & $\Z$    & 0 & $H^{5}(K(\Z,3))$ & $H^{6}(K(\Z,3))$ \\
    \hline
     & 0 & 1 & 2 & 3 & 4 & 5 & 6
\end{tabular}.
\]
 
 Now we use the spectral sequence to calculate  $H^{5}(K(\Z,3))$ and 
 $H^{6}(K(\Z,3))$. 
 Since every other row is 0, all of the differentials on page 2 
 are trivial and page 3 is the same as page 2. To fully analyze 
 page three we need to use the multiplicative
 structure (\ref{Derivations}). Let $x$ be a generator of
 $H^{3}(K(\Z,3))$,
 and consider the map 
 \[
 H^{0}(K(\Z,3);H^{2}(\C P^{\infty})) \xrightarrow{\partial}
 H^{3}(K(\Z,3);H^{0}(\C P^{\infty})).
 \]
 We need to understand the image of $z = z \cdot 1$. Upon inspection
 we see that the only differential which can affect $E_{\infty}^{0,2}$ or
 $E_{\infty}^{3,0}$ is the differential on page $3$. Furthermore, we
 know that $H^{n}(*) = 0$ for all $n$, and so $E_{\infty}^{p,q} = 0$
 whenever either $p$ or $q$ is non-zero.
 Thus $E_{\infty}^{0,2}$ and $E_{\infty}^{3,0}$ are both
 zero. This can only happen if $\partial(z)$ is a generator (and so we
 choose the sign of $x$ so that $\partial(z) = x$).
 
 Similarly, we can use the Leibintz
 property of the differentials to calculate 
 \[
 H^{0}(K(\Z,3),H^{4}(\C P^{\infty})) \xrightarrow{\partial}  
 H^{3}(K(\Z,3),H^{2}(\C P^{\infty})).
 \]
 Indeed, the generator of the left term is $z^{2}$, and
 $\partial(z^{2}) = \partial(z) z + z \partial(z) = x \cdot z + z 
 \cdot x$. Since $z$ has even degree, the cup product
  of $x$ and $z$ is commutative, and we get
 $z \cdot x + z\cdot x = 2 z \cdot x$.
 In particular the differential is multiplication by 2. We deduce that
 $E_{4}^{0,4}$ is zero (since the differential is injective.
 Furthermore, the differential 
 \[
 \Z/2\Z \cong E_{3}^{3,2} / \text{Im} \partial \xrightarrow{\partial} 
 H^{6}(K(\Z,3))
 \]
 must be an isomorphism; indeed everything on the $p+q = 5$ and $p+q =
 6$ diagonals must eventually die, and this is the last differential
 with a chance to kill either. We conclude that 
 \[
 H^{6}(K(\Z,3)) \cong \Z/2\Z.
 \]
 Similarly, $E_{3}^{5,0} = H^{5}(K(\Z,3))$ must die too. The only 
 differential with a chance to kill it is 
 \[
 0 = E_{5}^{0,4} \xrightarrow{\partial} H^{5}(K(\Z,3)),
 \]
 and so it must be zero.

%----------------------------------------------------------------------------
%----------------------------------------------------------------------------
\subsection{General Strategy}

 Let $1 \to A \to B \to C \to 1$ be an exact sequence of
 groups, and assume the further technical condition that $A$
 is in the center of $B$ (then the action of $/pi_{1}(K(C,1)) = C$ on
 $H_{*}(K(A,1);G)$ is trivial. 
 Then there is a fibration
 \[
 K(A,1) \to K(B,1) \to K(C,1).
 \]
 
 See \cite[Chapter 1, Section 1, Exercise 3]{SSS}). 
 This together with pathspace fibrations give a general first attempt 
 at computing homology and cohomology of $K(\G,n)$'s.
 
%**************************************************************************** 
 \section{Homotopy groups of spheres - $\pi_{4}(S^{3})$}
 This is one of the more interesting applications of spectral
 sequences. A good reference for more general facts about homotopy
 groups of spheres is \cite{rave}. Here we will use a spectral sequence to 
 calculate $\pi_{4}(S^{3})$. \\
 
 \label{sphere}
 $S^{3}$ has a CW-structure consisting of one 0-cell and 
 one 3-cell. We can kill off all higher homotopy groups by attaching 
 higher dimensional cells, forming a $K(\Z,3)$, and giving a
 fibration $\xymatrix{S^{3} \ar@{^{(}->}[r] & K(\Z,3)}$
 \[
 \xymatrix{
 F \ar[r]& S^{3} \ar[d]\\
 & K(\Z,3)
 } .
 \]
 From the long exact sequence of a fibration (\ref{LES}), we deduce that
 \[
 \pi_{i}(F) = \left\{\begin{array}{ll}
 \pi_{i}(S^{3}) & \text{if $i \geq 4$ } \\
0 & \text{ $i \leq 3$}  \end{array}\right. .
 \]
 The Hurewicz theorem (\ref{Hurewicz}), says that
 the first non-trivial homology and homotopy groups are isomorphic, so
 we conclude that
 \[
 \pi_{4}(S^{3}) \cong \pi_{4}(F) \cong H_{4}(F).
 \]
 We want to use a spectral sequence to compute $H^{5}(F)$ and
 $H^{4}(F)$. We can then
 use universal coefficients to compute $H_{4}(F)$.
 Indeed, using the additional data that $H_{i}(F)$ is 
 a finitely generated abelian group \cite[Chapter 1, Section 1]{SSS}, 
 we write $H_{4}(F) \cong \Z^{r} \oplus T$ (where $T$ is the finite
 and torsion). Universal Coefficients for cohomology (\ref{ucoco}) 
 gives the exact sequence
 \[
 1 \to \text{Ext}(H_{i-1}(F),\Z) \to H^{i}(F) \to
 \text{Hom}(H_{i}(F),\Z) \to 1.
 \]
 Letting $i = 4$ and recalling that $H_{3}(F) = 0$, we get that
 $\text{Hom}(H_{4}(F),\Z) = \Z^{r}$ is equal to the free part of
 $H^{4}(F)$, 
 and thus the ranks of $H_{4}(F)$ and $H^{4}(F)$ are equal. Letting $i
 = 5$ we get $H^{5}(F) \cong \text{Ext}(H_{4}(F),\Z) \oplus \text{Hom}
 (H_{5}(F),\Z)$. But $\text{Ext}(H_{4}(F),\Z) = T$. Thus we can deduce
 the free part of $H_{4}(F)$ from $H^{4}(F)$ and the torsion part from
 $H^{5}(F)$.

 To proceed we take one
 further fibration; by \cite[4.3]{HAT} we also have a fibration
 \[
 \Omega(K(\Z,3)) \to F \to S^{3}.
 \]
 From the long exact sequence of a pathspace fibration
 (\ref{pathspace}), we wee that 
 $\Omega(K(\Z,3))$ is a $K(\Z,2)$, which has the
 homotopy type of (and is thus homotopy equivalent to) $\C P^{\infty}$. 
 
 
 We can calculate (directly from the cell complexes!) that 
 \[
 H^{*}(S^{3}) \cong \Z[x]/(x^{2}), \text{degree } x = 3.
 \]
 and 
  \[
 H^{*}(\C P^{\infty}) \cong \Z[z], \text{degree } \alpha = 2.
 \]
 Setting $E^{p,q}_{2} := H^{p}(S^{3},H^{q}(\C P^{2}))$, we conclude that page two is
 \[
 \begin{tabular}{c|cccccc}
 5   & 0 & 0 & 0 & 0 & 0&0\\
 4   & $\Z$ & 0 & 0 & $\Z$ & 0&0\\
 3   & 0    & 0 & 0 & 0 & 0        &0\\
 2   & $\Z$    & 0 & 0 & $\Z$ & 0&0         \\
 1   & 0    & 0 & 0 & 0 & 0 &0       \\
 0   & $\Z$ & 0 & 0 & $\Z$ & 0 &0  \\
     \hline
      & 0 & 1 & 2 & 3 & 4 &5 \\
 \end{tabular}.
 \]
Rewriting page two with generators (for example, from
(\ref{Derivations}) the generator of $H^{3}(S^{3},
H^{2}(\C P^{2}))$ is just the cup product $z \cdot x$)
  \[
 \begin{tabular}{c|ccccccc}
 6   & $z^{3}$ & 0 & 0 & $z^{3}x$ & 0 &0 &0\\
 5   & 0 & 0 & 0 & 0 & 0 &0&0\\
 4   & $z^{2}$ & 0 & 0 & $z^{2}x$ & 0 &0&0\\
 3   & 0    & 0 & 0 & 0 & 0        &0&0\\
 2   & $z$    & 0 & 0 & $zx$ & 0    & 0 &0   \\
 1   & 0    & 0 & 0 & 0 & 0     &0   &0\\
 0   & $1$ & 0 & 0 & $x$ & 0  &0&0\\
     \hline
      & 0 & 1 & 2 & 3 & 4 &5 &6\\
 \end{tabular}
 \]
 the formal argument comes through. Nothing happens on page two, so page 
 three is identical.  We know by Hurewicz (\ref{Hurewicz})) that 
 $H_{2}(F)$ and $H_{3}(F)$ are 0, and by the universal coefficients 
 theorem for cohomology (\ref{ucoco}) $H^{2}(F)$ and $H^{3}(F) = 0$
 too.
 Thus $E_{\infty}^{2,0}$ and $E_{\infty}^{0,3}$ must both be 0, and
 the differential on page three from the (2,0) spot to the (0,3) spot 
 must be an isomorphism; $\partial(z) = x$.
 
 Using that $\partial$ is a derivation, we reason that
 $\partial(z^{2}) = 2zx$. Thus the differential from (0,4) to (3,2) is
 injective, and thus $E_{\infty}^{0,4} = 0$. We conclude that
 $H^{4}(F) = 0$. Similarly, $\partial(z^{p}) = pz\cdot x$, and so for 
 $i$ even the $p + q = i$ 
 diagonal is empty. We conclude that $H^{i}(F) = 0$ for $i$ even.
 
 Next consider the filtration
 \[
 H^{5}(F) \supset F^{5}_{0} \supset F^{5}_{1} \supset F^{5}_{2} \supset
 F^{5}_{3} \supset F^{5}_{4} \supset F^{5}_{5}
 \supset F^{5}_{6} = 0.
 \]
 Observe that $E_{\infty}^{p,5-p} = 0$ for $i = 0\ldots 2$, 
  $H^{5}(F) = F^{5}_{0} = \cdots F^{5}_{3}$ and 
 $F^{5}_{4} = F^{5}_{5} =  0$. Thus $F^{5}_{3} / F^{5}_{4} = F^{5}_{3} 
 = \Z/2\Z$. We conclude that $E_{\infty}^{3,2} = \Z/2\Z$ and 
 $E_{\infty}^{p,5-p} = 0$ if $p \neq 3$.
 With similar reasoning along the $p + q = i$ ($i$ odd) diagonal we conclude that 
 \[
 H^{i}(F) = \left\{\begin{array}{ll}
\Z/\left((i-1)/2\right)\Z & \text{$i$ odd } \\
0 & \text{$i$ even}  \end{array}\right.
 \]

 We have that $H^{4}(F) = 0$, and so the free part of $H_{4}(F)$ is zero.
 Also $H^{5}(F) = \Z/2\Z$, and so the torsion part of $H_{4}(F)$ is
 $\Z/2\Z$. We conclude that $H_{4}(F) = \pi_{4}(F) = \Z/2\Z$. 

%****************************************************************************
 \section{Grothendieck Spectral Sequence}
 
 Let $\mathscr{A}, \mathscr{B},$ and $\mathscr{C}$ be abelian categories, 
 and let $\mathscr{A} \xrightarrow{F} \mathscr{B} \xrightarrow{G}
 \mathscr{C}$ be two additive left exact functors such that the composition is 
 also left exact. Suppose our abelian categories have nice enough
 properties for right derived functors $R^{i}F$, $R^{i}G$ and
 $R^{i}(G\circ F)$ to exist, and suppose also that if $I$ is injective
 in $\mathscr{A}$, then $F(I)$ is $G$-acyclic. (See \cite[XX, section 9]{Lang}
 for these definitions, but don't worry much about them. They are just
 the most general technical conditions necessary for the following question
 to even make sense.) Once can ask the following\\
 
 \textbf{Question}: Is there any relation between $R^{p+q}(G \circ F)$
 and $R^{q}G \circ R^{p}F$?\\
  
  The answer to this loaded question is a resounding ``YES!'', and
  again comes from a spectral sequence. Indeed, in \cite[XX,
  Theorem 9.6]{Lang} we find the following:\\
  
  \begin{theorem} For each $A$ in $\mathscr{A}$, there is a spectral
  sequence $\{E_{r}(A)\}$ such that 
  \[
  E_{2}^{p,q}(A) = R^{p}G(R^{q}F(A))
  \]
  such that $E_{r}^{p,q}$ `converges' (in a similar sense as above,
  involving filtrations and extensions) to $R^{p+q}(G\circ F)(A)$.
  \end{theorem}
  
  Most of the spectral sequences we know and love arise as special
  cases of the Grothendieck Spectral Sequence, often as derived 
  functors of $M\otimes -$ or $\Hom(-,C)$ (the spectral sequence 
  described in (\ref{fibration}) arises from $\Hom$ on a suitable category of
  cellular complexes). A fun exercise for the 
  interested reader would be to derive the spectral sequences which
  appear `in nature' as instances of the Grothendieck spectral
  sequence. See also \cite[Chapter 12, Section 4]{user} for more on
  this.\\
  
  As demonstrated in the examples above, we can squeeze a lot of
  information out of the spectral sequence from purely formal
  considerations. The following lemma can be proved purely
  homologically (i.e. for a Grothendieck Spectral Sequence in an
  arbitrary abelian category), and nicely compactifies a few purely
  formal arguments one uses repeatedly. Setting  $L^{p+q} := 
  R^{p}G(R^{q}F(A))$ we get the following\\

  \begin{lemma} There is an exact sequence
  \[\label{lemma}
  0 \to E_{2}^{1,0} \to L^{1} \to E_{2}^{0,1} \xrightarrow{\partial}
  E_{2}^{2,0} \to \ker \left( L^{2} \to E_{2}^{0,2} \right)  \to
  E_{2}^{1,1} \to E_{2}^{3,0}.
  \]
  \end{lemma}

  The proof of this is purely formal and requires only minimal
  manipulations of exact sequences. In the next section we apply this 
  to group cohomology and observe as a corollary the
  inflation-restriction exact sequence.
  
  \remark Unfortunately, not all spectral sequences come from 
  this construction; as we will see in the appendix the cohomology of 
  the total complex of a double complex is the limit of a spectral
  sequence starting on page 0. 

%****************************************************************************
\section{Group Cohomology and the Hochschild-Serre Spectral
 Sequence}
 
 The Grothendieck Spectral Sequence gives us a clue as to how one
 might apply spectral sequences to other areas, Algebraic Geometry or 
 Number Theory. An amazing (to the author) fact is that spectral
 sequences can play a significant role in understanding and finding
 integer solutions to polynomial equations. Indeed, in \cite{PSS},
 spectral sequences are used as a key step in their classification of
 coprime integer solutions to the equation $x^2+y^3=z^7$.\\  
  
 Let $G$ be a group and let $M$ be a $G$-module (an abelian group
 equipped with an action of $G$). Consider  the \textbf{fixed-point functor} 
 \[
 M^{G}:= \{m \in M | gm = m \text{ for all } g \in G  \}.
 \] 
 An easy computation shows that this is left-exact, i.e. if 
 \[
 1 \to M' \to M \to M'' \to 1
 \]
 is exact, then 
 \[
 1 \to M'^{G} \to M^{G} \to M''^{G}
 \] 
 is also exact. It is a standard fact that the category of $G$-modules
 has `enough injectives' and satisfies the conditions necessary for
 right-derived functors of $-^{G}$ to exist. We call the right derived
 functor $H^{n}(G,A) := R^{i}(-^{G})(A)$ and call this `group
 cohomology'. The canonical reference for these facts is \cite{Serre}.\\

 Consider a short exact sequence of groups
 \[
 1 \to H \to G \to G/H \to 1.
 \]
 One hopes that there is a relation between the fixed point functors
 $-^{H}$, $-^{G}$, and $-^{G/H}$, and indeed, here we have a composition of functors
 waiting to be exploited!
 
 Let $M$ be a $G$-module. Then $M$ is also an $H$-module, and $H$ acts
 trivially on $M^{H}$; thus $M^{H}$ is a $G/H$-module. Furthermore,
 $(M^{H})^{G/H} = M^{G}$. This is the crucial observation, because now
 the composition of the functors
 \[
  \{G-\text{modules}\} \xrightarrow{-^{H}} \{G/H-\text{modules}\}
  \xrightarrow{-^{G/H}} 
   \mathscr{A} 
 \]
 is 
 \[
  \{G-\text{modules}\} \xrightarrow{-^{G}}
  \mathscr{A} 
 \]
 If $G$ and $H$ are finite abelian groups, then $\mathscr{A}$ is the category
 of abelian groups. In full generality the best one can do is
 $\mathscr{A} = \{$ pointed sets $\}$, for if $G$ is non-abelian,
 while $M^{G}$ is still an abelian group, the cohomology no longer has
 any group structure, only a canonical element.
  \david{Changed} There are many non-trivial results in number theory,
 co-prime integer solutions to $x^{2} + y^{3} = z^{7}$ for instance, whose 
 resolution entices one to care about the category of pointed sets.
 
 Now we turn the crank of the Grothendieck Spectral Sequence, setting
 \[
 E_{2}^{p,q} := H^{p}(G/H,H^{q}(H,A)),
 \]
 and conclude that it `converges' to $H^{p+q}(G,A)$.\\
 
 \remark The lemma from section (\ref{lemma}) amounts to the exactness of 
 
 \[
 \xymatrix{
 0 \ar[r] & H^{1}(G/H,A^{H}) \ar[r] & H^{1}(G,H) \ar[r] &
 H^{1}(H,A)^{G/H} \\
 \ar[r] & H^{2}(G/H,A^{H}) \ar[r] & \ker \left( H^{2}(G,A) \to
 H^{2}(H,A)\right) \ar[r] & H^{1}(G/H, H^{1}(H,A))\\
 \ar[r]& H^{3}(G/H,A^{H}). &&
 }
 \]
 This is a more general version of the \emph{inflation-restriction
 exact sequence}, which plays a role in group cohomology similar to
 the role of the long exact sequence of a fibration in topology.
 
 \remark Another application of the spectral sequence of a
 fibration is to develop group cohomology using twisted $K(G,1)$ spaces. 


%****************************************************************************

 \section{Appendix}
 Here I talk about some of the more abstract theory behind spectral
 sequences.\\
 
 
 Let $C^{\bullet}$ be a chain complex
 
 \[\xymatrix{
      \ldots \ar[r]^{\partial} & C_{d+1} \ar[r]^{\partial} & C_{d}
      \ar[r]^{\partial}& C_{d-1} \ar[r]^{\partial}& \ldots
 }\]
 whose homology we want to compute. For instance, we care about the cohomology 
 ring $H^{*}(X)$ of a topological space $X$. Often our chain complex
 has some extra 
 structure which makes things easier to compute. 
 Suppose, for example, that each $C_{d}$ was in fact a graded ring
 \[
 C_{d} = \bigoplus_{i = 1}^{\infty} C_{d,i},
 \]
 and that the boundary maps $\partial$ respect the grading:
 \[
 \partial(C_{d,i}) \subset C_{d-1,i}
 \]
 
 Here, understanding the grading and the homology of each graded part 
 allows us to recover the homology of the entire complex. Indeed, the 
 homology of the larger complex is just the sum of the homology of
 each graded part:
 \[
 h(C^{\bullet}) = \bigoplus_{i = 1}^{\infty} h(C_{i}^{\bullet}).
 \]
 In practice, we can't really compute the homology of all the graded
 parts, so we need more `extra structure'. \\

%---------------------------------------------------------------------------- 
 \subsection{Filtrations} 
 
 Consider the following more common situation: for each $d$ we have a 
 filtration
 \[
 0 = C_{d,0} \subset C_{d,1} \subset \ldots \subset C_{d,n} \subset \ldots
 \subset  C_{d},
 \] 
 such that the boundary maps respect the grading:
 \[
 \partial(C_{d,i}) \subset C_{d-1,i}.
 \]
 We can set 
 \[
 E^{0}_{d,i} := C_{d,i}/C_{d,i-1},
 \]
 \[
 C'_{d} := \bigoplus_{i=1}^{\infty}E_{d,i}^{0}.
 \]
 The boundary maps on $C_{d,i}$ induce maps 
 \[
  C_{d,i}/C_{d,i-1} = E^{0}_{d,i} \xrightarrow{\partial} E^{0}_{d-1,i}=
  C_{d-1,i}/C_{d-1,i-1}
 \]
 (since $C_{d,i-1}$ lands inside $C_{d-1,i-1}$). Starting with a
 filtration, we have stumbled upon a graded complex; we call this the
 \textbf{associated graded complex} of our filtration. Taking
 homology, we set 
 \[
 E^{1}_{d,i}:= \frac{\ker \left(E^{0}_{d,i} \xrightarrow{\partial}
 E^{0}_{d-1,i}\right)}{\im \left(E^{0}_{d+1,i} \xrightarrow{\partial}
 E^{0}_{d,i}\right)}
 \]
 and make the observation that $E^{1}_{d,i}$ is the relative homology
 group 
 \[
 H^{d}(C_{p},C_{p-1}).
 \]
 While the relative homology is related to the homology we originally 
 set out to calculate, they are not equal.\\ 
 
 Intuitively, the discrepancy
 occurs because the relative homology doesn't notice that sometimes
 the boundary maps move things down too many levels. For instance, it 
 may happen that 
 \[
 \partial(C_{d,i}) \subset C_{d-1,i-1},
 \]
 and the relative homology simply doesn't record this, while the homology does.

%----------------------------------------------------------------------------
 \subsection{Spectral Sequences}
 We begin by constructing a map
 \[
 E^{1}_{d,i} \xrightarrow{\partial} E^{1}_{d-1,i-1}.
 \]
 which measures the `discrepancy' of relative homology.\\
 
 Consider an $x \in C_{d,i}$. If $x \in \ker \left(E^{0}_{d,i} 
 \xrightarrow{\partial} E^{0}_{d-1,i}\right)$, then actually $x$ is in
 the kernel of the map
 \[
 C_{d,i} \xrightarrow{\partial} C_{d-1,i}/C_{d-1,i-1},
 \]
 i.e. $\partial(x) \in C_{d-1,i-1}$. This induces the desired map 
 $E^{1}_{d,i} \xrightarrow{\partial} E^{1}_{d-1,i-1}$, and further
 computations show that this in fact gives a graded chain complex
 \[
 0 \to \bigoplus_{i} E^{1}_{d+1,i+1}\to \bigoplus_{i} E^{1}_{d,i} 
 \to \bigoplus_{i} E^{1}_{d-1,i-1} \to \ldots.
 \]
 We can take homology and set
 \[
 E^{2}_{d,i}:= \frac{\ker \left(E^{1}_{d,i} \xrightarrow{\partial}
 E^{1}_{d-1,i-1}\right)}{\im \left(E^{1}_{d+1,i+1} \xrightarrow{\partial}
 E^{1}_{d,i}\right)}.
 \]
 Think of this as a second order approximation to $H^{*}$. Also, to
 make sense of this we should define 
 \[
 E^{m}_{d,i} := 0
 \]
 wherever $d$ or $i$ hasn't been defined yet. \\
 
 We think of this as only a second order approximation because, as
 before, we haven't taken into consideration chains $x \in C_{d,i}$
 which go down two levels, i.e such that
 \[
 \partial(x) \in C_{d-1,i-2}.
 \]
 Of course, we can iterate this process; the same arguments give a
 graded complex
 \[
 0 \to \bigoplus_{i} E^{2}_{d+1,i+2}\to \bigoplus_{i} E^{2}_{d,i} 
 \to \bigoplus_{i} E^{2}_{d-1,i-2} \to \ldots
 \]
 and more homology groups 
 \[
 E^{3}_{d,i}:= \frac{\ker \left(E^{2}_{d,i} \xrightarrow{\partial}
 E^{2}_{d-1,i-2}\right)}{\im \left(E^{2}_{d+1,i+2} \xrightarrow{\partial}
 E^{2}_{d,i}\right)}.
 \]
 More generally, we get 
 \[ 
 0 \to \bigoplus_{i} E^{r}_{d+1,i+r-1}\to \bigoplus_{i} E^{r}_{d,i} 
 \to \bigoplus_{i} E^{r}_{d-1,i-r+1} \to \ldots
 \]
 and
  \[
 E^{r+1}_{d,i}:= \frac{\ker \left(E^{r}_{d,i} \xrightarrow{\partial}
 E^{r}_{d-1,i-r+1}\right)}{\im \left(E^{r}_{d+1,i+r-1} \xrightarrow{\partial}
 E^{r}_{d,i}\right)}.
 \]
 We can take a limit (upon inspection we see that all maps eventually 
 become trivial and each individual homology group stabilizes) and set
 \[
 E^{\infty}_{d,i} := E^{r}_{d,i}
 \]
 for $r$ large enough that $E^{m}_{d,i}$ stabilizes.
 
%----------------------------------------------------------------------------
 \subsection{Comparison Theorems}
 Now we would like to understand how the graded ring 
 \[
 H^{*}(C^{\bullet})
 \]
 and our approximation, the doubly graded complex
 \[
 E(C^{\bullet}) := \bigoplus E^{\infty}_{d,i}
 \]
 are related. What is the best we can say? \\
 
 In our setup above, we have maps (inclusions even) of \emph{chain complexes}
 \[
 \xymatrix{
 \ldots \ar[r]&\ar[r]\ar[d] C_{d+1,i-1}& \ar[r]\ar[d] C_{d,i-1} &\ar[r]\ar[d]
 C_{d-1,i-1} & \ldots \\
  \ldots \ar[r]&\ar[r]\ar[d] C_{d+1,i}& \ar[r]\ar[d] C_{d,i} &\ar[r]\ar[d]
 C_{d-1,i} & \ldots \\
  \ldots \ar[r]&\ar[r] C_{d+1}& \ar[r] C_{d} &\ar[r]
 C_{d-1} & \ldots
 }
 \]
 or more compactly
  \[
 C^{\bullet}_{i-1} \to C^{\bullet}_{i} \to C^{\bullet}.
 \] 
 This induces maps
   \[
 H(C^{\bullet}_{i-1}) \to H(C^{\bullet}_{i}) \to H(C^{\bullet}),
 \] 
 and if we set 
 \[
 F_{i}(H^{*}) := \ker H(C^{\bullet}_{i}) \to H(C^{\bullet})
 \]
 we get another filtration
 \[
 0 \subset F_{1}(H^{*}) \subset F_{2}(H^{*}) \subset \ldots \subset
 F_{i}(H^{*}) \subset \ldots \subset H^{*}.
 \]\null
 
 Finally, set
 \[
 E^{\infty}_{d} := \bigoplus_{i}E^{\infty}_{i,d-i}
 \]
 and
 \[
 E = \bigoplus E^{\infty}_{d}.
 \]
 Then the best we can do is the following\\

 \begin{theorem} We have an isomorphism of graded rings
 \[
 \bigoplus E^{\infty}_{d} \cong \bigoplus F_{d}(H^{*})/F_{d-1}(H^{*}).
 \]
 \end{theorem}
 
 This gives a lot of partial information about $H^{*}$, but not full
 information. For example, the filtrations 
 \[
 0 \subset \Z/2\Z \subset \Z/4\Z
 \]
 and 
 \[
 0 \subset \Z/2\Z \subset \Z/2\Z \oplus \Z/2\Z
 \]
 look the same under the lens of this theorem. What we really have
 determined is $H^{*}$ up to extensions. 

%----------------------------------------------------------------------------
 \subsection{Example: Cohomology of a CW-Complex}
 We return now to the main example of the text. Consider a fibration
 $F \to X \xrightarrow{f} B$, with $B$ a CW-complex. Setting
 $X^{p} := f^{-1}(B^{p})$, we get a filtration
 \[
 X_{0} \subset X_{1} \subset X_{2} \subset \ldots \subset X_{i}
 \subset \ldots \subset X
 \]
 We set $C_{n}:=H^{n}(X)$ and $C_{n,p} := H^{n}(X_{p})$. The inclusions
 \[\xymatrix{
 X_{i-1} \ar@{^{(}->}[r] &X_{i} \ar@{^{(}->}[r]&X
 }
 \]
 induce maps 
 \[
 H^{*}(X) \to H^{*}(X_{i}) \to H^{*}(X_{i-1})
 \]
 and thus, setting 
 \[
 F_{i}(X):= \ker   \left( H^{*}(X) \to H^{*}(X_{i}) \right),
 \]
 a filtration
 \[
 0 \subset F_{1}(X) \subset F_{2}(X) \subset \ldots \subset F_{i}(X)
 \subset \ldots \subset H^{*}(X).
 \]
 The relative homology groups $E_{1}^{d,i}$ are the relative groups
 \[
 E_{1}^{d,i} = H^{d}(X_{i},X_{i-1}),
 \]
 and the homology of this gives us page 2 
 of the spectral sequence of section (\ref{fibration}).

 
%**************************************************************************** 
%\section*{Acknowledgments}



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%****************************************************************************

\end{document}
