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spectral sequence of a filtered complex

Idea

The spectral sequence of a filtered chain complex

$$\cdots \hookrightarrow F_{p-1}{C}_{•}\hookrightarrow F_p{C}_{•}\hookrightarrow \cdots \hookrightarrow C_{•}$$\cdots \hookrightarrow F_{p-1}C_\bullet \hookrightarrow F_p C_\bullet \hookrightarrow \cdots \hookrightarrow C_\bullet

is a tool for computing the chain homology of $C_{•}$ from the chain homologies of the associated graded objects

$$G_{p}C≔F_p{C}_{•}/F_{p-1}{C}_{•};$$G_p C \coloneqq F_p C_\bullet/F_{p-1}C_\bullet \,;

which is in general simpler.

The sequence asymptotes to the homology of $C_{•}$ by approximating cycles and boundaries of $C$ by their ”$r$-approximation”: an $r$-almost cycle is a chain in filtering degree $p$ whose differential vanishes only up to terms that are $r$ steps lower in filtering degree, and an $r$-almost boundary in filtering degree $p$ is a cycle that is the differential of a chain which may be (only) up to $r$-degrees higher in filtering degree. The corresponding $r$-almost homology of $C_{•}$ in filtering degree $p$ is the term $E_{p,•}^r$ of the spectral sequence.

If the filtering is bounded then $r$-almost homology for sufficiently large $r$ (”$\mathrm{\infty }$-almost homology”) is the clearly the genuine homology, and so the spectral sequence converges to the correct homology. But the point is that typically it reaches the correct value already at some low finite degree $r$ (it “collapses”), and so allows one to obtain the genuine homology from some finite $r$-almost homology.

One may also regard the spectral sequence of a filtered complex as a tool for organizing data derivable from the families of long exact sequence in homology

$$\cdots \to H_q(F_{p-1}{C}_{•})\to H_q(F_p{C}_{•})\to H_q(F_p{C}_{•}/F_{q-1}{C}_{•})\to H_{q-1}(F_{p-1}{C}_{•})\to \cdots$$\cdots \to H_q (F_{p-1} C_\bullet) \to H_q (F_p C_\bullet) \to H_q(F_p C_\bullet / F_{q-1} C_\bullet) \to H_{q-1}(F_{p-1} C_\bullet) \to \cdots

which are induced by the short exact sequences

$$0\to F_{p-1}{C}_q\hookrightarrow F_p{C}_q\to F_p{C}_q/F_{p-1}{C}_q\to 0$$0 \to F_{p-1}C_q \hookrightarrow F_p C_q \to F_p C_q / F_{p-1} C_q \to 0

coming from the filtering.

Definition

We give the definition

• via relative homology

and

• via exact couples.

Via relative homology

Let $R$ be a ring and write $𝒜=R$Mod for its category of modules.

Let

$$\cdots \hookrightarrow F_{p-1}{C}_{•}\hookrightarrow F_p{C}_{•}\hookrightarrow \cdots \hookrightarrow C_{•}$$\cdots \hookrightarrow F_{p-1}C_\bullet \hookrightarrow F_p C_\bullet \hookrightarrow \cdots \hookrightarrow C_\bullet

be a filtered chain complex in $𝒜$, with associated graded complex denoted $G_{•}{C}_{•}$.

Remark

In more detail this means that

1. $[\cdots \stackrel{{\partial }_n}{\to }{C}_n\stackrel{{\partial }_{n-1}}{\to }]C_{n-1}\to \cdots ]$ is a chain complex, hence $\{C_n\}$ are objects in $𝒜$ ($R$-modules) and $\{{\partial }_n\}$ are morphisms (module homomorphisms) with ${\partial }_n\circ {\partial }_{n+1}=0$;

2. For each $n\in \mathbb{Z}$ there is a filtering $F_{•}{C}_n$ on $C_n$ and all these filterings are compatible with the differentials in that

   $$\partial (F_p{C}_n)\subset F_p{C}_{n-1}$$\partial(F_p C_n) \subset F_p C_{n-1}

3. The grading associated to the filtering is such that the $p$-graded elements are those in the quotient

   $$G_p{C}_n≔\frac{F_p{C}_n}{F_{p-1}{C}_n}.$$G_p C_n \coloneqq \frac{F_p C_n}{ F_{p-1} C_n} \,.

   Since the differentials respect the grading we have chain complexes $G_p{C}_{•}$ in each filtering degree $p$.

We use element-notation in the following as if $𝒜$ were a category of modules.

$r$-Almost cycles and boundaries

Definition

Given a filtered chain complex $F_{•}{C}_{•}$ as above we say for all $r,p,q\in \mathbb{Z}$ that

1. $G_p{C}_{p+q}$ is the module of $(p,q)$-chains or of $(p+q)$-chains in filtering degree $p$;

2. $\begin{array}{rl}{Z}_{p,q}^r& ≔\{c\in G_p{C}_{p+q}\mid \partial c=0\mathrm{mod}{F}_{p-r}{C}_{•}\}\\ & =\{c\in F_p{C}_{p+q}\mid \partial (c)\in F_{p-r}{C}_{p+q-1}\}/F_{p-1}{C}_{p+q}\end{array}$

   is the module of $r$-almost $(p,q)$-cycles (the $(p+q)$-chains whose differential vanishes modulo terms of filtering degree $p-r$);

3. $B_{p,q}^r≔\partial (F_{p+r-1}{C}_{p+q+1}),$

   is the module of $r$-almost $(p,q)$-boundaries.

Similarly we set

$$Z_{p,q}^{\mathrm{\infty }}≔\{c\in F_p{C}_{p+q}\mid \partial c=0\}/F_{p-1}{C}_{p+q}=Z(G_p{C}_{p+q})$$Z^\infty_{p,q} \coloneqq \{c \in F_p C_{p + q} | \partial c = 0 \}/F_{p-1}C_{p+q} = Z(G_p C_{p+q})

$$B_{p,q}^{\mathrm{\infty }}=\partial (F_p{C}_{p+q+1}).$$B^\infty_{p,q} = \partial( F_p C_{p+q+1} ) \,.

From this definition we immediately have that the differentials $\partial :C_{p+q}\to C_{p+q-1}$ restrict to the $r$-almost cycles as follows:

Proposition

The differentials of $C_{•}$ restrict on $r$-almost cycles to morphisms of the form

$${\partial }^r:Z_{p,q}^r\to Z_{p-r,q+r-1}^r.$$\partial^r \colon Z^r_{p,q} \to Z^r_{p-r, q+r-1} \,.

These are still differentials: ${\partial }^2=0$.

Proof

By the very definition of $Z_{p,q}^r$ it consists of elements in filtering degree $p$ on which $\partial$ decreases the filtering degree to $p-r$. Also by definition of differential on a chain complex, $\partial$ decreases the actual degree $p+q$ by one. This explains that $\partial$ restricted to $Z_{p,q}^r$ lands in $Z_{p-r,q+r-1}^{•}$.

Now the image constists indeed of actual boundaries, not just $r$-almost boundaries. But since actual boundaries are in particular $r$-almost boundaries, we may take the codomain to be $Z_{p-r,q+r-1}^r$.

Proposition

We have a sequence of canonical inclusions

$$B_{p,q}^0\hookrightarrow B_{p,q}^1\hookrightarrow \cdots B_{p,q}^{\mathrm{\infty }}\hookrightarrow Z_{p,q}^{\mathrm{\infty }}\hookrightarrow \cdots \hookrightarrow Z_{p,q}^1\hookrightarrow Z_{p,q}^0.$$B^0_{p,q} \hookrightarrow B^1_{p,q} \hookrightarrow \cdots B^\infty_{p,q} \hookrightarrow Z^\infty_{p,q} \hookrightarrow \cdots \hookrightarrow Z^1_{p,q} \hookrightarrow Z^0_{p,q} \,.

Proposition

The $(r+1)$-almost cycles are the ${\partial }^r$-kernel inside the $r$-almost cycles:

$$Z_{p,q}^{r+1}=\ker (Z_{p,q}^r\stackrel{{\partial }^r}{\to }{Z}_{p-r,q+r-1}^r).$$Z^{r+1}_{p,q} = ker( Z^r_{p,q} \stackrel{\partial^r}{\to} Z^r_{p-r, q+r-1} ) \,.

Proof

An element $c\in F_p{C}_{p+q}$ represents

1. an element in $Z_{p,q}^r$ if $\partial c\in F_{p-r}{C}_{p+q-1}$

2. an element in $Z_{p,q}^{r+1}$ if even $\partial c\in F_{p-r-1}{C}_{p+q-1}\hookrightarrow F_{p-r}{C}_{p+q-1}$.

The second condition is equivalent to $\partial c$ representing the 0-element in the quotient $F_{p-r}{C}_{p+q-1}/F_{p-r-1}{C}_{p+q-1}$. But this is in turn equivalent to $\partial c$ being 0 in $Z_{p-r,q+r-1}^r\subset F_{p-r}{C}_{p+q-1}/F_{p-r-1}{C}_{p+q-1}$.

$r$-Almost homology groups: the spectral sequence

Let $F_{•}{C}_{•}$ be a filtered chain complex as above.

Definition

For $r,p,q\in \mathbb{Z}$ define the $r$-almost $(p,q)$-chain homology of the filtered complex to be the quotient of the $r$-almost $(p,q)$-cycles by the $r$-almost $(p,q)$-boundaries, def. 1:

$$\begin{array}{rl}{E}_{p,q}^r& ≔\frac{Z_{p,q}^r}{B_{p,q}^r}\\ & =\frac{\{x\in F_p{C}_{p+q}\mid \partial x\in F_{p-r}{C}_{p+q-1}\}}{\partial (F_{p+r-1}{C}_{p+q+1})\oplus F_{p-1}{C}_{p+q}}\end{array}$$\begin{aligned} E^r_{p,q} & \coloneqq \frac{Z^r_{p,q}}{B^r_{p,q}} \\ & = \frac{ \left\{ x \in F_p C_{p+q} \,|\, \partial x \in F_{p-r} C_{p+q-1} \right\} } { \partial( F_{p+r-1} C_{p+q+1} ) \oplus F_{p-1} C_{p+q} } \end{aligned}

By prop. 1 the differentials of $C_{•}$ restrict on the $r$-almost homology groups to maps

$${\partial }^r:E_{p,q}^r\to E_{p-r,q+r-1}^r.$$\partial^r : E^r_{p,q} \to E^r_{p-r, q + r - 1} \,.

Proposition

Definition 2 indeed gives a spectral sequence in that $E_{•,•}^{r+1}$ is indeed the ${\partial }^r$-chain homology of $E_{•,•}^r$, i.e.

$$E_{p,q}^{r+1}=\frac{\ker ({\partial }_r:E_{p,q}^r\to E_{p-r,q+r-1}^r)}{\mathrm{im}({\partial }_r:E_{p+r,q-r+1}^r\to E_{p,q}^r)}.$$E^{r+1}_{p,q} = \frac{ ker(\partial_r : E^r_{p,q} \to E^r_{p-r, q+r-1}) }{ im( \partial_r : E^r_{p+r, q-r+1} \to E^r_{p,q} ) } \,.

Proof

By prop. 3.

Via exact couples

The whole of the spectral sequence can be defined as the spectral sequence of the exact couple

$$D\stackrel{\phi }{\to }D\to E\to D$$D\overset{\varphi}{\to} D \to E \to D

where

• $D≔{⨁}_i{H}^{•}(F_i)$

• and where $\phi$ is the cohomology morphism induced by the inclusion of chain complexes $F_i\to F_{i+1}$

• and $E≔{⨁}_i{H}^{•}(F_i/F_{i-1})$ is the total cohomology of the associated bigraded complex.

At every stage we have a new family of long exact sequences

Properties

In low-degree pages

We characterize the value of the spectral sequence $E_{p,q}^r$, def. 2 for low values of $r$ and, below in prop. \ref{ConvergingToGenuineHomology}, for $r\to \mathrm{\infty }$.

Proposition

We have

• $E_{p,q}^0=G_p{C}_{p+q}=F_p{C}_{p+q}/F_{p-1}{C}_{p+q}$

  is the associated p-graded piece of $C_{p+q}$;

• $E_{p,q}^1=H_{p+q}(G_p{C}_{•})$

  is the chain homology of the associated p-graded complex $G_p{C}_{•}$.

Proof

For $r=0$ def. 2 restricts to

$$E_{p,q}^0=\frac{F_p{C}_{p+q}}{F_{p-1}{C}_{p+q}}=G_p{C}_{p+q}$$E^0_{p,q} = \frac{ F_p C_{p+q}}{F_{p-1} C_{p+q}} = G_p C_{p+q}

because for $c\in F_p{C}_{p+q}$ we automatically also have $\partial c\in F_p{C}_{p+q}$ since the differential respects the filtering degree by assumption.

For $r=1$ def. 2 gives

$$E_{p,q}^1=\frac{\{c\in G_p{C}_{p+q}\mid \partial c=0\in G_p{C}_{p+q}\}}{\partial (F_p{C}_{p+q})}=H_{p+q}(G_p{C}_{•}).$$E^1_{p,q} = \frac{\{c \in G_p C_{p+q} | \partial c = 0 \in G_p C_{p+q}\} }{\partial(F_p C_{p+q})} = H_{p+q} (G_p C_\bullet) \,.

Remark

There is, in general, a decisive difference between the homology of the associated graded complex $H_{p+q}(G_p{C}_{•})$ and the associated graded piece of the genuine homology $G_p{H}_{p+q}(C_{•})$: in the former the differentials of cycles are required to vanish only up to terms in lower degree, but in the latter they are required to vanish genuinely. The latter expression is instead the value of the spectral sequence for $r\to \mathrm{\infty }$, see prop. \ref{ConvergingToGenuineHomology} below.

Convergence

General definitions

Definition

Let $\{E_{p,q}^r{\}}_{r,p,q}$ be a spectral sequence such that for each $p,q$ there is $r(p,q)$ such that for all $r\ge r(p,q)$ we have

$$E_{p,q}^{r\ge r(p,q)}\simeq E_{p,q}^{r(p,q)}.$$E^{r \geq r(p,q)}_{p,q} \simeq E^{r(p,q)}_{p,q} \,.

Then one says that

1. the bigraded object

   $$E^{\mathrm{\infty }}≔\{E_{p,q}^{\mathrm{\infty }}{\}}_{p,q}≔\{E_{p,q}^{r(p,q)}{\}}_{p,q}$$E^\infty \coloneqq \{E^\infty_{p,q}\}_{p,q} \coloneqq \{ E^{r(p,q)}_{p,q} \}_{p,q}

   is the limit term of the spectral sequence;

• the spectral sequence abuts to $E^{\mathrm{\infty }}$.

Example

If for a spectral sequence there is $r_s$ such that all differentials on pages after $r_s$ vanish, ${\partial }^{r\ge r_s}=0$, then $\{E^{r_s}{\}}_{p,q}$ is limit term for the spectral sequence. One says in this cases that the spectral sequence collapses at $r_s$.

Proof

By the defining relation

$$E_{p,q}^{r+1}\simeq \ker ({\partial }_{p-r,q+r-1}^r)/\mathrm{im}({\partial }_{p,q}^r)=E_{\mathrm{pq}}^r$$E^{r+1}_{p,q} \simeq ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) = E^r_{pq}

the spectral sequence becomes constant in $r$ from $r_s$ on if all the differentials vanish, so that $\ker ({\partial }_{p,q}^r)=E_{p,q}^r$ for all $p,q$.

Example

If for a spectral sequence $\{E_{p,q}^r{\}}_{r,p,q}$ there is $r_s\ge 2$ such that the $r_s$th page is concentrated in a single row or a single column, then the the spectral sequence degenerates on this pages, example 1, hence this page is a limit term, def. 3. One says in this case that the spectral sequence collapses on this page.

Proof

For $r\ge 2$ the differentials of the spectral sequence

$${\partial }^r:E_{p,q}^r\to E_{p-r,q+r-1}^r$$\partial^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}

have domain and codomain necessarily in different rows an columns (while for $r=1$ both are in the same row and for $r=0$ both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.

Definition

A spectral sequence $\{E_{p,q}^r{\}}_{r,p,q}$ is said to converge to a graded object $H_{•}$ with filtering $F_{•}{H}_{•}$, traditionally denoted

$$E_{p,q}^r\Rightarrow H_{•},$$E^r_{p,q} \Rightarrow H_\bullet \,,

if the associated graded complex $\{G_p{H}_{p+q}{\}}_{p,q}≔\{F_p{H}_{p+q}/F_{p-1}{H}_{p+q}\}$ of $H$ is the limit term of $E$, def. 3:

$$E_{p,q}^{\mathrm{\infty }}\simeq G_p{H}_{p+q}{\forall }_{p,q}.$$E^\infty_{p,q} \simeq G_p H_{p+q} \;\;\;\;\;\;\; \forall_{p,q} \,.

Remark

In practice spectral sequences are often referred to via their first non-trivial page, often also the page at which it collapses, def. 2, oftne the second page. Then one often uses notation such as

$$E_{p,q}^2\Rightarrow H_{•}$$E^2_{p,q} \Rightarrow H_\bullet

to be read as “There is a spectral sequence whose second page is as shown on the left and which converges to a filtered object as shown on the right.”

Definition

A spectral sequence $\{E_{p,q}^r\}$ is called a bounded spectral sequence if for all $n,r\in \mathbb{Z}$ the number of non-vanishing terms of the form $E_{k,n-k}^r$ is finite.

Example

A spectral sequence $\{E_{p,q}^r\}$ is called

• a first quadrant spectral sequence if all terms except possibly for $p,q\ge 0$ vanish;

• a third quadrant spectral sequence** if all terms except possibly for $p,q\le 0$ vanish.

Such spectral sequences are bounded, def. 5.

Proposition

A bounded spectral sequence, def. 5, has a limit term, def. 3.

Proof

First notice that if a spectral sequence has at most $N$ non-vanishing terms of total degree $n$ on page $r$, then all the following pages have at most at these positions non-vanishing terms, too, since these are the homologies of the previous terms.

Therefore for a bounded spectral sequence for each $n$ there is $L(n)\in \mathbb{Z}$ such that $E_{p,n-p}^r=0$ for all $p\le L(n)$ and all $r$. Similarly there is $T(n)\in \mathbb{Z}$ such $E_{n-q,q}^r=0$ for all $q\le T(n)$ and all $r$.

We claim then that the limit term of the bounded spectral sequence is in position $(p,q)$ given by the value $E_{p,q}^r$ for

$$r>\max (p-L(p+q-1),q+1-L(p+q+1)).$$r \gt max( p-L(p+q-1), q + 1 - L(p+q+1) ) \,.

This is because for such $r$ we have

1. $E_{p-r,q+r-1}^r=0$ because $p-r<L(p+q-1)$, and hence the kernel $\ker ({\partial }_{p-r,q+r-1}^r)=0$ vanishes;

2. $E_{p+r,q-r+1}^r=0$ because $q-r+1<T(p+q+1)$, and hence the image $\mathrm{im}({\partial }_{p,q}^r)=0$ vanishes.

Therefore

$$\begin{array}{rl}{E}_{p,q}^{r+1}& =\ker ({\partial }_{p-r,q+r-1}^r)/\mathrm{im}({\partial }_{p,q}^r)\\ & \simeq E_{p,q}^r/0\\ & \simeq E_{p,q}^r\end{array}.$$\begin{aligned} E^{r+1}_{p,q} &= ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) \\ & \simeq E^r_{p,q}/0 \\ & \simeq E^r_{p,q} \end{aligned} \,.

Convergence of bounded filtrations

Definition

A filtration $F_{•}{C}_{•}$ on a chain complex $C_{•}$ is called a bounded filtration if for all $n\in \mathbb{Z}$ there is $L(n),T(n)\in \mathbb{Z}$ such that

$$F_{p<S(n)}{C}_n=0F_{p>T(n)}{C}_n\simeq C_n.$$F_{p \lt S(n)}C_n = 0 \;\;\;\;\; F_{p \gt T(n)}C_n \simeq C_n \,.

Proposition

The spectral sequence of a filtered complex, def. 6, has a limit term, def. 3.

Proof

The spectral sequence of a filtered complex $F_{•}{C}_{•}$ is by def. 2 at $E_{p,q}^r$ a quotient of a subobject of $F_p{C}_{p+q}$. By def. 6 therefore there are for each $n,r\in \mathbb{Z}$ finitely many non-vanishing terms of the form $E_{p,n-p}^r$. Therefore the spectral sequence is bounded, def. 5 and hence has a limit term by prop. 6.

Example

If $X\in$ Top is a CW complex with cell filtration $X_0\hookrightarrow X_1\hookrightarrow \cdots \hookrightarrow X$, then the induced filtering

$$F_p{C}_{•}(X)≔C_{•}(X_n)$$F_p C_\bullet(X) \coloneqq C_\bullet(X_n)

on its singular chain complex $C_{•}(X)$ yields a first-quadrant spectral sequence, example 3. Therefore it has a limit term.

Before saying what that the spectral sequence of a filtered complex converges to the homology of that complex, we need to be careful about what the filtering is on that homology:

Definition

For $F_{•}{C}_{•}$ a filtered complex, write for $p\in \mathbb{Z}$

$$F_p{H}_{•}(C)≔\mathrm{image}(H_{•}(F_p{C}_{•})\to H_{•}(C_{•})).$$F_p H_\bullet(C) \coloneqq image( H_\bullet(F_p C_\bullet) \to H_\bullet(C_\bullet) ) \,.

This defines a filtering $F_{•}{H}_{•}(C)$ of the homology, regarded as a graded object.

Proposition

If the spectral sequence of a filtered complex $F_{•}{C}_{•}$, def. 6 has a limit term, def. 3, then it converges, def. 4, to the chain homology of this complex:

$$E_{p,q}^r\Rightarrow H_{•}(C).$$E^r_{p,q} \Rightarrow H_\bullet(C) \,.

Hence for each $p,q$ there is $r(p,q)$ such that

$$E_{p,q}^{r\ge r(p,q)}\simeq G_p{H}_{p+q}(C),$$E^{r \geq r(p,q)}_{p,q} \simeq G_p H_{p+q}(C) \,,

with the filtering on the right as in def. 7.

Proof

By assumption, there is for each $p,q$ an $r(p,q)$ such that for all $r\ge r(p,q)$ the $r$-almost cycles and $r$-almost boundaries, def. 1, in $F_p{C}_{p+q}$ are the ordinary cycles and boundaries. Therefore for $r\ge r(p,q)$ def. 2 gives $E_{p,q}^r\simeq G_p{H}_{p+q}(C)$.

Via exact couples

It is instructive to note that in the $n$th derived exact couple ${\phi }^{n}D\to E_{(n)}\to {\phi }^{n}D\to$, the hidden part ${\phi }^{n}D$ is the submodule $D_{(n)}$ of ${⨁}_{i}H(F_{i+n})$, as it meets $H(F_{i+n})$ representable by elements of $F_i$; that is, we may sensibly call it

$$F_i{D}_{(n)}=\frac{\ker (d)\cap F_i}{F_i\cap {\mathrm{dF}}_{i+n}}.$$F_{i} D_{(n)} = \frac{\ker(d)\cap F_i}{F_i\cap dF_{i+n}}.

Separating the grades, the exactness of the couple at $E_{(n)}$ then says

$$F_i{H}^j(F_{i+n})\to F_{i+1}{H}^j(F_{i+n+1})\to E_{(n)}^{i\dots }\to F_i{H}^{j+1}(F_{i+n})\to F_{i+1}{H}^{j+1}(F_{i+n+1})$$F_{i} H^j(F_{i+n})\to F_{i+1}H^j(F_{i+n+1}) \to E_{(n)}^{i\dots} \to F_i H^{j+1}(F_{i+n}) \to F_{i+1}H^{j+1}(F_{i+n+1})

One can see this as converging (if it sensibly converges) to either a subquotient of $F_i$ or to a submodule $F_i{H}^{•}(C)<H^{•}(C)$. Taking the latter interpretation, we hope to find in the limiting case exact sequences

$$F_i{H}^j(C)\to F_{i+1}{H}^j(C)\to E_{(\mathrm{\infty })}^{i\dots }\to F_i{H}^{j+1}(C)\to F_{i+1}{H}^{j+1}(C).$$F_i H^j(C)\to F_{i+1} H^j(C) \to E_{(\infty)}^{i\dots} \to F_i H^{j+1}(C) \to F_{i+1} H^{j+1}(C).

At this stage one can check that the morphisms $F_i{H}^j(C)\to F_{i+1}{H}^j(C)$ are indeed definable, and in fact injective, so that whatever $E_{(\mathrm{\infty })}$ should be, the morphism $E_{(\mathrm{\infty })}^{i\dots }\to F_i{H}^{j+1}$ is null; that is, our long-exact sequence breaks up into the short exact sequences

$$0\to F_i{H}^j(C)\to F_{i+1}{H}^j(C)\to E_{(\mathrm{\infty })}^{i\dots }\to 0.$$0\to F_i H^j(C) \to F_{i+1} H^j(C)\to E_{(\infty)}^{i\dots} \to 0 .

In summary, if the spectral sequence $E_{(n)}$ converges in a sensible way to the correct thing $E_{(\mathrm{\infty })}$, then that correct thing is also the associated graded module of the filtration of $H^{•}(C)$ induced by the filtration of $C$.

Examples

2-term filtering

The special case where the filtering has just length one is that where we simply have a sub-complex $C_{•}^{(1)}\hookrightarrow C_{•}$ and want to compute the homology of $C_{•}$ from that of $C_{•}^{(1)}$ and $C_{•}/C_{•}^{(1)}$.

This case is easily solved by elementary means and it serves as an instructive blueprint for the general case.

Proposition

Given a sub-chain complex $C_{•}^{(1)}\hookrightarrow C_{•}$, consider the following constructions

1. Consider the short exact sequence

   $$0\to C_{•}^{(1)}\to C_{•}\to C_{•}/C_{•}^{(1)}\to 0$$0 \to C^{(1)}_\bullet \to C_\bullet \to C_\bullet/C^{(1)}_\bullet \to 0

2. Its long exact sequence in homology contains the connecting homomorphism

   $$\delta :H_{•}(C_{•}/C_{•}^{(1)})\to H_{•-1}(C_{•}^{(1)}).$$\delta : H_\bullet(C_\bullet/C^{(1)}_\bullet) \to H_{\bullet-1}(C^{(1)}_\bullet) \,.

   Define

   • $G_1{H}_{•}≔\ker \delta$

   • $G_2{H}_{•}≔\mathrm{coker}\delta$.

Then $H_{•}(C_{•})$ is sits in the short exact sequence

$$0\to G_0{H}_{•}\to H_{•}(C_{•})\to G_1{H}_{•}\to 0.$$0 \to G_0 H_\bullet \to H_\bullet(C_\bullet) \to G_1 H_\bullet \to 0 \,.

(…)

Homology of a tensor product of chain complexes

Consider two chain complexes $C_{•},C{\prime }_{•}$ of vector spaces over a field $k$, both in non-negative degree.

Their tensor product of chain complexes is

$$C_{•}\otimes C{\prime }_{•}={\oplus }_{i+j=k}{C}_i\otimes C_j$$C_\bullet \otimes C'_\bullet = \oplus_{i+j = k} C_i \otimes C_j

with differential on homogenous elements

$$\partial (\alpha \otimes \beta )=(\partial \alpha )\otimes \beta +(-1{)}^{\mathrm{deg}\alpha }\alpha \otimes \partial \beta .$$\partial (\alpha \otimes\beta) = (\partial \alpha) \otimes \beta + (-1)^{deg \alpha} \alpha \otimes \partial \beta \,.

We may compute the chain homology of $C\otimes C\prime$ by a filter spectral sequence as follows.

Define a filtration on $C\otimes C\prime$ by

$$F_p(C\otimes C\prime {)}_k≔{\oplus }_{i\le p}{C}_i\otimes C_{k-1}.$$F_p(C \otimes C')_{k} \coloneqq \oplus_{i \leq p} C_i \otimes C_{k-1} \,.

This means that the associated graded object is simply

$$E_{p,q}^0=G_p(C\otimes C\prime {)}_{p+q}=C_p\otimes C{\prime }_q.$$E^0_{p,q} = G_p (C \otimes C')_{p+q} = C_p \otimes C'_q \,.

The differential on this is ${\partial }_{r=0}=(-1{)}^p{\mathrm{id}}_C\otimes \partial \prime$. Hence the universal coefficient theorem gives

$$E_{p,q}^1=C_p\otimes H_q(C\prime ).$$E^1_{p,q} = C_p \otimes H_q(C') \,.

The next differential is ${\partial }_1=\partial \otimes {\mathrm{id}}_{C\prime }$ Since $k$ is assumed to be a field we have thus

$$E_{p,q}^2=H_p(C_{•}\otimes H_q(C{\prime }_{•}))=H_p(C)\otimes H_q(C\prime ).$$E^2_{p,q} = H_p(C_\bullet \otimes H_q(C'_\bullet)) = H_p(C)\otimes H_q(C') \,.

Therefore every element in $E_{p,q}^2$ is represented by a tensor product of a $C$-cycle with a $C\prime$-cycle and is hence itself a $(C\otimes C\prime )$-cycle. Since the differentials in the spectral sequence all come from the differential on $C\otimes C\prime$, this means that all higher differentials vanish, and so the sequence collapses on the $E^2$-page.

The convergence of the spectral sequence to the the homology of $C\otimes C\prime$ thus says that this is given by

$$H_k(C\otimes C\prime )\simeq {\oplus }_{i+j=k}{H}_i(C)\otimes H_j(C\prime ).$$H_{k}(C \otimes C') \simeq \oplus_{i+j = k} H_i(C) \otimes H_j(C') \,.

Homology of the total complex of a double complex

The total complex of a double complex is naturally filtered either by either row-degree of column-degree. The corresponding filtering spectral sequence converges under good conditions to the homology of the total complex. See at spectral sequence of a double complex.

Singular homology of a CW-complex

Let $X\in \mathrm{Top}$ be a CW-complex equipped explicitly with the structure of a filtered topological space $X^0\hookrightarrow \cdots \hookrightarrow X^n\hookrightarrow \cdots X$. This induces on the singular homology complex $C_{•}(X)$ the structure of a filtered chain complex by

$$F_q{C}_{•}(X)≔C_{•}(X^p).$$F_q C_\bullet(X) \coloneqq C_\bullet(X^p) \,.

We discuss how the corresponding spectral sequence shows that the singular homology of $X$ coincides with the cellular homology of the filtering.

The associated graded object is

$$G_p{C}_{p+q}(X)=E_{p,q}^0=C_{p+q}(X^p)/C_{p+q}(X^{p-1}).$$G_p C_{p+q}(X) = E^0_{p,q} = C_{p+q}(X^p)/C_{p+q}(X^{p-1}) \,.

Proposition

The chain homology of the associated graded chain complex is the relative homology

$$E_{p,q}^1=H_{p+q}(X^p\mid X^{p-1}).$$E^1_{p,q} = H_{p+q}(X^p | X^{p-1}) \,.

Now by assumption that $X^{•}$ is the cell decomposition of a cell complex we have

$$H_{p+q}(X^p\mid X^{p+1})\simeq \{\begin{array}{cc}\mathbb{Z}[\mathrm{pCells}(X)]& q=0\\ 0& \mathrm{otherwise}\end{array}.$$H_{p+q}(X^p | X^{p+1}) \simeq \left\{ \array{ \mathbb{Z}[pCells(X)] & q = 0 \\ 0 & otherwise } \right. \,.

The chain homology of

$$\partial :H_p(X^p,X^{p-1})\to H_{p-1}(X^{p-1},X^{p-2})$$\partial \colon H_p(X^p, X^{p-1}) \to H_{p-1}(X^{p-1}, X^{p-2})

is the cellular chain homology $H_p^{\mathrm{cell}}(X)$. One finds that

$$E_{p,q}^2=\{\begin{array}{cc}{H}_p^{\mathrm{cell}}(X)& q=0\\ 0& \mathrm{otherwise}\end{array}.$$E^2_{p,q} = \left\{ \array{ H_p^{cell}(X) & q = 0 \\ 0 & otherwise } \right. \,.

Since this is concentrated in the $q=0$-row all higher-$r$ differentials vanish.

Hence $H_p(X)\simeq H_p^{\mathrm{cell}}(X)$.

Remark

The generalization of this argument from ordinary homology to generalized homology is given by the Atiyah–Hirzebruch spectral sequence.

Serre’s spectral sequence of a fibration

We indicate the Leray-Serre spectral sequence of a Serre fibration as a special case of the filtering spectral sequence. For more discussion see there.

$$\begin{array}{ccc}A& \to & X\\ \downarrow & & \downarrow \\ *& \to & B\end{array}$$\begin{matrix} A & \to & X \\ \downarrow & & \downarrow \\ * & \to & B \end{matrix}

be a Serre fibration of pointed topological spaces in which $B$ is a connected CW-complex. Then the $k$-skeleta of $B$ naturally give filtered-space structures to both $B$ and $X$:

$$\begin{array}{ccccc}{X}_{(k)}& \to & X_{(k+1)}& \to & X\\ \downarrow & & \downarrow & & \downarrow \\ B_k& \to & B_{k+1}& \to & B\end{array}$$\begin{matrix} X_{(k)} & \to & X_{(k+1)} & \to & X \\ \downarrow & & \downarrow & &\downarrow \\ B_{k} & \to & B_{k+1} & \to & B \end{matrix}

and in turn induce filtrations of the singular chain complex of $X$.

The homology Serre spectral sequence for the fibration is essentially that of this filtered complex.

It is straight-forward to show that the pair $(X_{k+1};X_k)$ is $k$-connected, and in particular the relative homology $H_i(X_{k+1};X_k)$ vanishes for $i\le k$; this ensures that the spectral sequence is 1st/3rd quadrant (And nota bene: this is also a handy way to remember what the bigrading actually is).

There is also an important result about the second page of this spectral sequence

Theorem

The page $E^{(2)}$ of the homology Serre spectral sequence is given by

$$E_{p,q}^{(2)}\simeq H_p(B,{\mathscr{H}}_q(A{\mid }_b))$$E^{(2)}_{p,q} \simeq H_p(B,\mathcal{H}_q(A|_{b}))

the homology of $B$ with coefficients in the local system defined by the action of ${\Pi }_1(B)$ on $H_q(A{\mid }_b)$. In the special case that $B$ is simply connected, these local systems are canonically equivalent to $H_q(A)$, the homology of the fiber over the basepoint.

References

A detailed account is in

• Jennifer Orlich, Spectral sequences and an application (pdf)

Lecture notes include

• Michael Hutchings, Introduction to spectral sequences (2011) (pdf)

and section 3 of

• Brandon Williams, Spectral sequences (pdf)

For further references see those listed at spectral sequence, for instance section 5 of

• Charles Weibel, An Introduction to Homological Algebra