Exact couple

In mathematics, an exact couple, due to William S. Massey (1952), is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple.

For the definition of an exact couple and the construction of a spectral sequence from it (which is immediate), see Spectral sequence § Spectral Sequence of an exact couple. For a basic example, see Bockstein spectral sequence. The present article covers additional materials.

Exact couple of a filtered complex

Let R be a ring, which is fixed throughout the discussion. Note if R is $\mathbb {Z}$, then modules over R are the same thing as abelian groups.

Each filtered chain complex of modules determines an exact couple, which in turn determines a spectral sequence, as follows. Let C be a chain complex graded by integers and suppose it is given an increasing filtration: for each integer p, there is an inclusion of complexes:

$F_{p-1}C\subset F_{p}C.$

From the filtration one can form the associated graded complex:

$\operatorname {gr} C=\bigoplus _{-\infty }^{\infty }F_{p}C/F_{p-1}C,$

which is doubly-graded and which is the zero-th page of the spectral sequence:

$E_{p,q}^{0}=(\operatorname {gr} C)_{p,q}=(F_{p}C/F_{p-1}C)_{p+q}.$

To get the first page, for each fixed p, we look at the short exact sequence of complexes:

$0\to F_{p-1}C\to F_{p}C\to (\operatorname {gr} C)_{p}\to 0$

from which we obtain a long exact sequence of homologies: (p is still fixed)

$\cdots \to H_{n}(F_{p-1}C){\overset {i}{\to }}H_{n}(F_{p}C){\overset {j}{\to }}H_{n}(\operatorname {gr} (C)_{p}){\overset {k}{\to }}H_{n-1}(F_{p-1}C)\to \cdots$

With the notation $D_{p,q}=H_{p+q}(F_{p}C),\,E_{p,q}^{1}=H_{p+q}(\operatorname {gr} (C)_{p})$, the above reads:

$\cdots \to D_{p-1,q+1}{\overset {i}{\to }}D_{p,q}{\overset {j}{\to }}E_{p,q}^{1}{\overset {k}{\to }}D_{p-1,q}\to \cdots ,$

which is precisely an exact couple and $E^{1}$ is a complex with the differential $d=j\circ k$. The derived couple of this exact couple gives the second page and we iterate. In the end, one obtains the complexes $E_{*,*}^{r}$ with the differential d:

$E_{p,q}^{r}{\overset {k}{\to }}D_{p-1,q}^{r}{\overset {{}^{r}j}{\to }}E_{p-r,q+r-1}^{r}.$

The next lemma gives a more explicit formula for the spectral sequence; in particular, it shows the spectral sequence constructed above is the same one in more traditional direct construction, in which one uses the formula below as definition (cf. Spectral sequence#The spectral sequence of a filtered complex).

Lemma — Let $A_{p}^{r}=\{c\in F_{p}C\mid d(c)\in F_{p-r}C\}$, which inherits $\mathbb {Z}$-grading from $F_{p}C$. Then for each p

$E_{p,*}^{r}\simeq {A_{p}^{r} \over d(A_{p+r-1}^{r-1})+A_{p-1}^{r-1}}.$

Sketch of proof: Remembering $d=j\circ k$, it is easy to see:

$Z^{r}=k^{-1}(\operatorname {im} i^{r}),\,B^{r}=j(\operatorname {ker} i^{r}),$

where they are viewed as subcomplexes of $E^{1}$.

We will write the bar for $F_{p}C\to F_{p}C/F_{p-1}C$. Now, if $[{\overline {x}}]\in Z_{p,q}^{r-1}\subset E_{p,q}^{1}$, then $k([{\overline {x}}])=i^{r-1}([y])$ for some $[y]\in D_{p-r,q+r-1}=H_{p+q-1}(F_{p}C)$. On the other hand, remembering k is a connecting homomorphism, $k([{\overline {x}}])=[d(x)]$ where x is a representative living in $(F_{p}C)_{p+q}$. Thus, we can write: $d(x)-i^{r-1}(y)=d(x')$ for some $x'\in F_{p-1}C$. Hence, $[{\overline {x}}]\in Z_{p}^{r}\Leftrightarrow x\in A_{p}^{r}$ modulo $F_{p-1}C$, yielding $Z_{p}^{r}\simeq (A_{p}^{r}+F_{p-1}C)/F_{p-1}C$.

Next, we note that a class in $\operatorname {ker} (i^{r-1}:H_{p+q}(F_{p}C)\to H_{p+q}(F_{p+r-1}C))$ is represented by a cycle x such that $x\in d(F_{p+r-1}C)$. Hence, since j is induced by ${\overline {\cdot }}$, $B_{p}^{r-1}=j(\operatorname {ker} i^{r-1})\simeq (d(A_{p+r-1}^{r-1})+F_{p-1}C)/F_{p-1}C$.

We conclude: since $A_{p}^{r}\cap F_{p-1}C=A_{p-1}^{r-1}$,

$E_{p,*}^{r}={Z_{p}^{r-1} \over B_{p}^{r-1}}\simeq {A_{p}^{r}+F_{p-1}C \over d(A_{p+r-1}^{r-1})+F_{p-1}C}\simeq {A_{p}^{r} \over d(A_{p+r-1}^{r-1})+A_{p-1}^{r-1}}.\qquad \square$

Theorem — If $C=\cup _{p}F_{p}C$ and for each n there is an integer $s(n)$ such that $F_{s(n)}C_{n}=0$, then the spectral sequence E^r converges to $H_{*}(C)$; that is, $E_{p,q}^{\infty }=F_{p}H_{p+q}(C)/F_{p-1}H_{p+q}(C)$.

Proof: See the last section of May. $\square$

Exact couple of a double complex

A double complex determines two exact couples; whence, the two spectral sequences, as follows. (Some authors call the two spectral sequences horizontal and vertical.) Let $K^{p,q}$ be a double complex. With the notation $G^{p}=\bigoplus _{i\geq p}K^{i,*}$, for each with fixed p, we have the exact sequence of cochain complexes:

$0\to G^{p+1}\to G^{p}\to K^{p,*}\to 0.$

Taking cohomology of it gives rise to an exact couple:

$\cdots \to D^{p,q}{\overset {j}{\to }}E_{1}^{p,q}{\overset {k}{\to }}\cdots$

By symmetry, that is, by switching first and second indexes, one also obtains the other exact couple.

Example: Serre spectral sequence

The Serre spectral sequence arises from a fibration:

$F\to E\to B.$

For the sake of transparency, we only consider the case when the spaces are CW complexes, F is connected and B is simply connected; the general case involves more technicality (namely, local coefficient system).

References

• May, J. Peter, A primer on spectral sequences (PDF)
• Massey, William S. (1952), "Exact couples in algebraic topology. I, II", Annals of Mathematics, Second Series, 56: 363–396, doi:10.2307/1969805, MR 0052770.
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge: Cambridge University Press, doi:10.1017/CBO9781139644136, ISBN 0-521-43500-5, MR 1269324