Exact couple

Short description: Algebraic topology

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 $\mathbb{\mathbb{Z}}$, 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:

$\mathrm{gr}C=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{⨁}}{F}_{p}C/F_{p-1}C,$

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

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

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

$0\to F_{p-1}C\to F_{p}C\to (\mathrm{gr}C{)}_p\to 0$

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

$\cdots \to H_n(F_{p-1}C)\stackrel{i}{\to }{H}_n(F_{p}C)\stackrel{j}{\to }{H}_n(\mathrm{gr}(C{)}_p)\stackrel{k}{\to }{H}_{n-1}(F_{p-1}C)\to \cdots$

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

$\cdots \to D_{p-1,q+1}\stackrel{i}{\to }{D}_{p,q}\stackrel{j}{\to }{E}_{p,q}^1\stackrel{k}{\to }{D}_{p-1,q}\to \cdots ,$

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:

$E_{p,q}^r\stackrel{k}{\to }{D}_{p-1,q}^r\stackrel{{}^{r}j}{\to }{E}_{p-r,q+r-1}^r.$

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).

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

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

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

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

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 $[\bar{x}]\in Z_{p,q}^{r-1}\subset E_{p,q}^1$, then $k([\bar{x}])=i^{r-1}([y])$ 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, $k([\bar{x}])=[d(x)]$ where x is a representative living in $(F_{p}C{)}_{p+q}$. Thus, we can write: $d(x)-i^{r-1}(y)=d(x^{\prime })$ for some $x^{\prime }\in F_{p-1}C$. Hence, $[\bar{x}]\in Z_p^r\iff x\in A_p^r$ 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 $\ker (i^{r-1}:H_{p+q}(F_{p}C)\to H_{p+q}(F_{p+r-1}C))$ is represented by a cycle x such that $x\in d(F_{p+r-1}C)$. Hence, since j is induced by $\bar{\cdot }$, $B_p^{r-1}=j(\ker i^{r-1})\simeq (d(A_{p+r-1}^{r-1})+F_{p-1}C)/F_{p-1}C$.

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

$E_{p,*}^r=\frac{Z_p^{r-1}}{B_p^{r-1}}\simeq \frac{A_p^r+F_{p-1}C}{d(A_{p+r-1}^{r-1})+F_{p-1}C}\simeq \frac{A_p^r}{d(A_{p+r-1}^{r-1})+A_{p-1}^{r-1}}.◻$

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, $E_{p,q}^{\mathrm{\infty }}=F_p{H}_{p+q}(C)/F_{p-1}{H}_{p+q}(C)$.

Proof: See the last section of May. $◻$

Exact couple of a double complex

A double complex determines two exact couples; whence, the two spectral sequences, are as follows. (Some authors call the two spectral sequences horizontal and vertical.) Let $K^{p,q}$ be a double complex. With the notation $G^p=\underset{i\ge p}{⨁}{K}^{i,*}$, 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:

$\cdots \to D^{p,q}\stackrel{j}{\to }{E}_1^{p,q}\stackrel{k}{\to }\cdots$

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, http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf 
• "Exact couples in algebraic topology. I, II", Annals of Mathematics, Second Series 56: 363–396, 1952, doi:10.2307/1969805 .
• An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge: Cambridge University Press, 1994, doi:10.1017/CBO9781139644136, ISBN 0-521-43500-5