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\title{Notes on algebraic topology}
\date{\today}

\begin{document}

\maketitle

\tableofcontents

\part{Spectral sequences}
\section{Background}
\begin{defn}
A graded $R$-module $M$ is an $R$-module which decomposes as a direct
sum
\begin{equation*}
\bigoplus_{p\in\Z} F_p M
\end{equation*}
of $R$-modules. A graded $R$-homomorphism $h:M\to N$ is an $R$-homomorphism which
decomposes into $h_p:F_pM\to F_pN$. 
\end{defn}

\begin{lem}
Suppose $M$ and $N$ are graded $R$-modules. Then $M\otimes N$ is a graded
$R$-module by
\begin{equation*}
(M\otimes_R N)_i\defeq \bigoplus_{p+q=i} F_pM\otimes_R F_qN.
\end{equation*}
\end{lem}

\begin{defn}
A graded algebra is a graded $R$-module $M$ for which there are linear mappings
$\varphi_{p,q}:F_pM\otimes_R F_qM\to F_{p+q}M$, i.e.~a graded $R$-homomorphism
$\varphi:M\otimes M\to M$, which is associative in the sense
that the diagram
\begin{equation*}
\begin{tikzcd}
M\otimes M\otimes M \arrow[r,"\varphi\otimes 1"] \arrow[d,swap,"1\otimes\varphi"] &
M\otimes M \arrow[d,"\varphi"] \\ M\otimes M \arrow[r,swap,"\varphi"] & M
\end{tikzcd}
\end{equation*}
commutes.
\end{defn}

\begin{eg}
Polynomials with coefficients in $R$ forms a graded algebra. Moreover, in the
polynomial ring $R[X]$, we find that $G_pR[X]\defeq F_pR[X]/F_{p-1}R[X]\cong R$.
Since those are free modules, we have that $R[X]\cong \bigoplus_p G_pR[X]$. 
\end{eg}

\section{Spectral sequences}

\subsection{Motivation from the long exact sequence of a pair}

Recall that a pair of spaces $(X,A)$ induces a long exact sequence of homology
groups
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r,"\partial_{n+1}"]
& H_n(A) \arrow[r,"i_n"]
& H_n(X) \arrow[r,"j_n"]
& H_n(X,A) \arrow[r,"\partial_n"]
& H_{n-1}(A) \arrow[r,"i_{n-1}"]
& \cdots
\end{tikzcd}
\end{equation*}
from the short exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & C_\ast(A) \arrow[r] & C_\ast(X) \arrow[r] & C_\ast(X,A) \arrow[r] & 0
\end{tikzcd}
\end{equation*}
of chain complexes, by means of the snake lemma. This long exact sequence helps
us to compute $H_n(X)$ in terms of $H_n(A)$ and $H_n(X,A)$, which may be easier
to determine. For instance, from the long exact sequence we obtain the short
exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & \mathrm{coker}(\partial_{n+1}) \arrow[r] & H_n(X) \arrow[r] & \mathrm{ker}(\partial_n) \arrow[r] & 0
\end{tikzcd}
\end{equation*}
and hence we have determined that $H_n(X)$ can be obtained as some element of the
group $\mathrm{Ext}(\mathrm{coker}(\partial_{n+1}),\mathrm{ker}(\partial_n))$.
In other words, $H_n(X)$ is a particular solution to an extension problem.

Note also that the long exact sequence of relative homology groups can be
presented as an exact triangle of graded $R$-homomorphisms:
\begin{equation*}
\begin{tikzcd}[column sep=0em]
\bigoplus_n H_n(C_\ast(A)) 
\arrow[rr,"i"] & & \bigoplus_n H_n(C_\ast(X)) \arrow[dl,"j"] \\
& \bigoplus_n H_n(C_\ast(X,A)) \arrow[ul,"\partial"]
\end{tikzcd}
\end{equation*}

The first idea of spectral sequences is to generalize the long exact sequence
of homology obtained from a pair of spaces, to an algebraic gadget obtained from
a filtration on a space, and mimic the derivation of determining the homology
group as a solution to an extension problem.

\begin{defn}
A filtration of a space X consists of a sequence 
\begin{equation*}
\cdots\subseteq X_p\subseteq X_{p+1}\subseteq\cdots
\end{equation*}
such that $X=\bigcup_p X_p$ and $\bigcap_p X_p=\varnothing$. A filtration of $X$ is said to be bounded, if
$X_p=\varnothing$ for $p$ sufficiently small, and $X_p=X$ for $X$ sufficiently
large.
\end{defn}

An important class of filtered spaces is that of CW-complexes, where the filtration
$X_p$ of $X$ is given by the $p$-skeleton of $X$. Another case is where
$X_p\defeq\varnothing$ for $p<0$, $X_0\defeq A$ and $X_p\defeq X$ for $p>0$; here
we recover the old theory of the topological pair.

\begin{defn}
Given a space $X$ with a filtration, we can form the staircase diagram
\begin{footnotesize}
\begin{equation*}
\begin{tikzcd}
& \vdots \arrow[d] & & \vdots \arrow[d] \\ 
\cdots \arrow[r]
& H_{n+1}(X_p) \arrow[r] \arrow[d]
& H_{n+1}(X_p,X_{p-1}) \arrow[r]
& H_n(X_{p-1}) \arrow[r] \arrow[d]
& H_n(X_{p-1},X_{p-2}) \arrow[r]
& \cdots \\
\cdots \arrow[r]
& H_{n+1}(X_{p+1}) \arrow[r] \arrow[d]
& H_{n+1}(X_{p+1},X_{p}) \arrow[r]
& H_n(X_{p}) \arrow[r] \arrow[d]
& H_n(X_{p},X_{p-1}) \arrow[r]
& \cdots \\
& \vdots & & \vdots
\end{tikzcd}
\end{equation*}%
\end{footnotesize}%
in which the familiar long exact sequence of the pairs $(X_p,X_{p-1})$ run
down like a staircase.
\end{defn}

\begin{defn}
Let $X$ be a space with a filtration. Then we obtain the exact couple
\begin{equation*}
\begin{tikzcd}
A \arrow[rr,"i"] & & A \arrow[dl,"j"] \\
& E \arrow[ul,"\partial"]
\end{tikzcd}
\end{equation*}
in which $A\defeq\bigoplus_{p,n} H_n(X_p)$, and $E\defeq\bigoplus_{p,n}H_n(X_p,X_{p-1})$. 
\end{defn}

We can come to such an exact couple from any filtered chain complex, which is
what we turn our attention to before continuing.

\subsection{The spectral sequence of a filtered complex}

\begin{defn}
A filtration of an $R$-module $M$ consists of a sequence
\begin{equation*}
\cdots\subseteq F_pM\subseteq F_{p+1}M\subseteq\cdots
\end{equation*}
of $R$-submodules of $M$, such that $M=\bigcup_p F_pM$ and $\bigcap_p F_pM=0$. 
A filtration of $R$ is said to be bounded if $F_pM=0$ for $p$ sufficiently
small and $F_pM=M$ for $p$ sufficiently large.
\end{defn}

\begin{defn}
Let $\{M,F_pM\}$ be a graded $R$-module. The associated graded module is defined
by $G_p M\defeq F_pM/F_{p-1}M$. We obtain a short exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & F_{p-1}M \arrow[r] & F_pM \arrow[r] & G_pM \arrow[r] & 0.
\end{tikzcd}
\end{equation*}
\end{defn}

\begin{rmk}
It would be nice if $F_pM\cong F_{p-1}M\oplus G_pM$, so that we can write
$M\cong\bigoplus_p G_pM$. Under what condition does this hold? This holds if
each $G_pM$ is a projective $R$-module, so under what conditions is this true?
\end{rmk}

\begin{defn}
A filtered chain complex is a chain complex $(C_\ast,\partial)$ together with a
filtration $\{F_pC_i\}$ of each $C_i$, such that the differential preserves the
filtration, i.e.~$\partial(F_pC_i)\subseteq F_p C_{i-1}$. 

A filtration of a chain complex is said to be bounded if it is bounded in each
dimension.
\end{defn}

Let $(F_pC_\ast,\partial)$ be a filtered chain complex. We have again our
short exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & F_{p-1} C_\ast \arrow[r] & F_p C_\ast \arrow[r] & G_p C_\ast \arrow[r] & 0
\end{tikzcd}
\end{equation*}
of chain complexes. This also gives us the long exact sequence on homology,
which we may express conveniently as the exact couple
\begin{equation*}
\begin{tikzcd}[column sep=0em]
\bigoplus_{p,q} H_{p+q}(F_pC_\ast) \arrow[rr,"i"] & & \bigoplus_{p,q} H_{p+q}(F_pC_\ast) \arrow[dl,"j"] \\
& \bigoplus_{p,q} H_{p+q}(G_p C_\ast) \arrow[ul,"k"]
\end{tikzcd}
\end{equation*}
consisting of graded $R$-homomorphisms (of which $k$ shifts in degree).

\begin{defn}
Consider an exact couple, i.e.~a commutative triangle
\begin{equation*}
\begin{tikzcd}
A \arrow[rr,"i"] & & A \arrow[dl,"j"] \\ & E \arrow[ul,"k"]
\end{tikzcd}
\end{equation*}
of $R$-modules, which is exact at every vertex. Taking $\partial^0\defeq j\circ k$,
we see that $(\partial^0)^2=0$ by exactness. We may now form the derived exact couple
\begin{equation*}
\begin{tikzcd}[column sep=0]
\mathrm{im}(i) \arrow[rr,"i'"] & & \mathrm{im}(i) \arrow[dl,"j'"] \\
& \frac{\mathrm{ker}(\partial)}{\mathrm{im}(\partial)} \arrow[ul,"k'"]
\end{tikzcd}
\end{equation*}
where
\begin{align*}
i'(i(a)) & \defeq i(i(a)) \\
j'(i(a)) & \defeq [j(a)] \\
k'([e]) & \defeq k(e)
\end{align*}
\end{defn}

\begin{rmk}
Since quotients commute with direct sums (both are colimits), it follows that
\begin{equation*}
E'\defeq \frac{\mathrm{ker}(\partial)}{\mathrm{im}(\partial)}
  \cong
\bigoplus_{p,q} \frac{\mathrm{ker}(\partial^0_{p,q})}{\mathrm{im}(\partial^0_{p,q+1})}
\end{equation*}
is a graded $R$-module. In other words, $E'$ is a direct sum of the homology
groups of the $p$-indexed family of chain complexes
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] & E_{p,q}^0 \arrow[r,"{\partial^0_{p,q}}"] & E_{p,q-1}^0 \arrow[r] & \cdots
\end{tikzcd}
\end{equation*}
It follows that $i'$, $j'$ and $k'$ are graded
whenever $i$, $j$ and $k$ are, where $k'$ shifts down in dimension the same way 
$k$ does.
\end{rmk}

\begin{comment}
\begin{defn}
We define
\begin{equation*}
E_{p,q}^0\defeq G_pC_{p+q}\defeq F_pC_{p+1}/F_{p-1}C_{p+q},
\end{equation*}

Since the differential preserves the filtration, we obtain from the differentials
well-defined $R$-homomorphisms functioning as the boundary maps in the chain complex

\end{defn}

\begin{defn}
The homology groups 
\begin{equation*}
E^1_{p,q}\defeq \mathrm{ker}(\partial^0_{p,q})/\mathrm{im}(\partial^0_{p,q+1})
\end{equation*}
form again a chain complex, with boundary maps $\partial^1_{p,q}:E^1_{p,q}\to
E^1_{p,q-1}$. Thus, this process may be repeated indefinitely.
\end{defn}
\end{comment}

\begin{comment}
\begin{lem}
Let $(C_\ast,\partial)$ be a filtered chain complex. Then there is a filtration
on the homology of $C_\ast$, given by
\begin{equation*}
F_pH_i(C_\ast)\defeq\{\alpha\in H_i(C_\ast)\mid \exists_{(x\in F_p C_i)}\,\alpha=[x]\}.
\end{equation*}
\end{lem}
\end{comment}

\subsection{Convergent spectral sequences}

\begin{defn}
A spectral sequence consists of
\begin{enumerate}
\item An $R$-module $E^r_{p,q}$ for each $p,q\in\Z$ and each $r\geq 0$.
\item Differentials $\partial_r:E^r_{p,q}\to E^r_{p-r,q+r-1}$ such that
$\partial_r^2=0$ and $E^{r+1}$ is the homology of $(E^r,\partial_r)$ 
\end{enumerate}
\end{defn}

\begin{defn}
A spectral sequence $\{E^r,\partial_r\}$ of $R$-modules is said to converge 
if for every $p,q\in\Z$, one has $\partial_r=0:E^r_{p,q}\to E^r_{p-r,q+r-1}$
for $r$ sufficiently large.
\end{defn}

\begin{rmk}
If a spectral sequence $\{E^r,\partial_r\}$ converges, then the $R$-module
$E^r_{p,q}$ is independent of $r$ for sufficiently large $r$. 
\end{rmk}

\begin{thm}
Let $(F_pC_\ast,\partial)$ be a filtered complex. Then we obtain a spectral
sequence $(E^r_{p,q},\partial^r)$ defined for $r\geq 0$, with
\begin{equation*}
E^1_{p,q}\defeq H_{p+q}(G_pC_\ast).
\end{equation*}
This is the spectral sequence of filtered complexes.
\end{thm}

\begin{thm}
If $(F_pC_\ast,\partial)$ is a bounded filtered complex, then the spectral
sequence converges to
\begin{equation*}
E^\infty_{p,q}\defeq G_pH_{p+q}(C_\ast).
\end{equation*}
\end{thm}

Let $X$ be a filtered space, and let our goal be to compute the $n$-th (co)homology
group $H_n(X)$. In general, this might be a complicated task. However, it might
be easier to compute the homologies of the subcomplex $C_\ast(X_p)$, and the quotient
complex $C_\ast(X)/C_\ast(X_p)$. From this, we obtain a short exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow[r]
& \mathrm{coker}(\delta) \arrow[r]
& H_\ast(X) \arrow[r]
& \mathrm{ker}(\delta) \arrow[r]
& 0
\end{tikzcd}
\end{equation*}

\subsection{The Serre spectral sequence}

The Serre spectral sequence relates the homology of a Serre fibration to the
homology of the fibers and the base. Thus, in some cases one can compute the
homology of the fibration in terms of the homology of the fibers and the base.

Let $\pi : X\to B$ be a fibration, with $B$ a path-connected CW-complex, and we
filter $X$ by the subspaces $X_p\defeq \pi^{-1}(B_p)$, in which $B_p$ is the
$p$-skeleton of $B$. 

\begin{lem}
The spectral sequence for homology with coefficients in $G$ associated to this
filtration of $X$ converges to $H_\ast(X;G)$.
\end{lem}

\begin{thm}
Let $F\to X\to B$ be a fibration with $B$ path-connected. If $\pi_1(B)$ acts
trivially on $H_\ast(F;G)$, then there is a spectral sequence $\{E^r_{p,q},\partial_r\}$
with:
\begin{enumerate}
%\item $\partial_r : E^r_{p,q}\to E^r_{p-r,q+r-1}$ and $E^{r+1}_{p,q}=\mathrm{ker}\,d_r/\mathrm{im}\,dr$. 
\item the stable terms $E^\infty_{p,n-p}$ are isomorphic to $F^p_n/F^{p-1}_n$ in
a filtration $0\subseteq F^0_n\subseteq\cdots\subseteq F^n_n=H_n(X;G)$ of ...
\item $E^2_{p,q}\cong H_p(B;H_q(F;G))$. 
\end{enumerate}
\end{thm}

\end{document}