Contents

# Contents

## Idea

Exact couples are a way to encode data that makes a spectral sequence, specially adapted to the case that the underlying filtering along which the spectral sequence proceeds is induced from a tower of homotopy fibers, such as a Postnikov tower or Adams tower (see also at Adams spectral sequence).

## Definition

### Exact couples

###### Definition

Given an abelian category $\mathcal{C}$, an exact couple in $\mathcal{C}$ is a cyclic long exact sequence of three morphisms among two objects of the form

$\cdots \stackrel{k}{\longrightarrow} E \overset{j}{\to} D \overset{\varphi}{\longrightarrow} D \overset{k}{\to} E \overset{j}{\longrightarrow} \cdots \,.$
###### Remark

This being cyclic, it is usually depicted as a triangle

$\array{ D && \stackrel{\varphi}{\longrightarrow} && D \\ & {}_{\mathllap{j}}\nwarrow && \swarrow_{\mathrlap{k}} \\ && E }$

The archetypical example from which this and the following definition draw their meaning is example below.

### Spectral sequences from exact couples

###### Definition

A cohomology spectral sequence $\{E_r^{p,q}, d_r\}$ is

1. a sequence $\{E_r^{\bullet,\bullet}\}$ $r \in \mathbb{Z}$, $r \geq 2$ of bigraded abelian groups;

2. a sequence of differentials $\{d_r \colon E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}$

such that

• $H_{r+1}^{\bullet,\bullet}$ is the cochain cohomology of $d_r$:, i.e. $E_{r+1}^{\bullet, \bullet} = H(E_r^{\bullet,\bullet},d_r)$.

Given a $\mathbb{Z}$-graded abelian group_ $C^\bullet$ equipped with a decreasing filtration

$C^\bullet \supset \cdots \supset F^s C^\bullet \supset F^{s+1} C^\bullet \supset \cdots \supset 0$

such that

$C^\bullet = \underset{s}{\cup} F^s C^\bullet \;\;\;\; and \;\;\;\; 0 = \underset{s}{\cap} F^s C^\bullet$

then the spectral sequence is said to converge to $C^\bullet$, denoted,

$E_2^{\bullet,\bullet} \Rightarrow C^\bullet$

if

1. in each bidegree $(s,t)$ the sequence $\{E_r^{s,t}\}_r$ eventually becomes constant on a group

$E_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t}$;

2. $E_\infty^{\bullet,\bullet}$ is the associated graded of the filtered $C^\bullet$ in that

$E_\infty^{s,t} \simeq F^s C^{s+t} / F^{s+1}C^{s+t}$.

The converging spectral sequence is called multiplicative if

1. $\{E_2^{\bullet,\bullet}\}$ is equipped with the structure of a bigraded object algebra;

2. $F^\bullet C^\bullet$ is equipped with the structure of a filtered graded algebra ($F^p C^k \cdot F^q C^l \subset F^{p+q} C^{k+l}$);

such that

1. each $d_{r}$ is a derivation with respect to the (induced) algebra structure on ${E_r^{\bullet,\bullet}}$, graded of degree 1 with respect to total degree;

2. the multiplication on $E_\infty^{\bullet,\bullet}$ is compatible with that on $C^\bullet$.

###### Definition

(derived exact couples)

An exact couple is three homomorphisms of abelian groups of the form

$\array{ D && \stackrel{g}{\longrightarrow} && D \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h}} \\ && E }$

such that the image of one is the kernel of the next.

$im(h) = ker(f)\,,\;\;\; im(f) = ker(g)\,, \;\;\; im(g) = ker(h) \,.$

Given an exact couple, then its derived exact couple is

$\array{ im(g) && \stackrel{g}{\longrightarrow} && im(g) \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h \circ g^{-1}}} \\ && H(E, h \circ f) } \,.$

Here and in the following we write $g^{-1}$ etc. for the operation of choosing a preimage under a given function $g$. In each case it is left implicit that the given expression is independent of which choice is made.

###### Proposition

(cohomological spectral sequence of an exact couple)

Given an exact couple, def. ,

$\array{ D_1 && \stackrel{g_1}{\longrightarrow} && D_1 \\ & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}} \\ && E_1 }$

its derived exact couple

$\array{ D_2 && \stackrel{g_2}{\longrightarrow} && D_2 \\ & {}_{\mathllap{f_2}}\nwarrow && \swarrow_{\mathrlap{h_2}} \\ && E_2 }$

is itself an exact couple. Accordingly there is induced a sequence of exact couples

$\array{ D_r && \stackrel{g_r}{\longrightarrow} && D_r \\ & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}} \\ && E_r } \,.$

If the abelian groups $D$ and $E$ are equipped with bigrading such that

$deg(f) = (0,0)\,,\;\;\;\; deg(g) = (-1,1)\,,\;\;\; deg(h) = (1,0)$

then $\{E_r^{\bullet,\bullet}, d_r\}$ with

\begin{aligned} d_r & \coloneqq h_r \circ f_r \\ & = h \circ g^{-r+1} \circ f \end{aligned}

is a cohomological spectral sequence, def. .

(As before in prop. , the notation $g^{-n}$ with $n \in \mathbb{N}$ denotes the function given by choosing, on representatives, a preimage under $g^n = \underset{n\;times}{\underbrace{g \circ \cdots \circ g \circ g}}$, with the implicit claim that all possible choices represent the same equivalence class.)

If for every bidegree $(s,t)$ there exists $R_{s,t} \gg 1$ such that for all $r \geq R_{s,t}$

1. $g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1}$;

2. $g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}$

then this spectral sequence converges to the inverse limit group

$G^\bullet \coloneqq \underset{}{\lim} \left( \cdots \stackrel{g}{\to} D^{s,\bullet-s} \stackrel{g}{\longrightarrow} D^{s-1, \bullet - s + 1} \stackrel{g}{\to} \cdots \right)$

filtered by

$F^p G^\bullet \coloneqq ker(G^\bullet \to D^{p-1, \bullet - p+1}) \,.$

(e.g. Kochmann 96, lemma 2.6.2)

###### Proof

We check the claimed form of the $E_\infty$-page:

Since $ker(h) = im(g)$ in the exact couple, the kernel

$ker(d_{r-1}) \coloneqq ker(h \circ g^{-r+2} \circ f)$

consists of those elements $x$ such that $g^{-r+2} (f(x)) = g(y)$, for some $y$, hence

$ker(d_{r-1})^{s,t} \simeq f^{-1}(g^{r-1}(D^{s+r-1,t-r+1})) \,.$

By assumption there is for each $(s,t)$ an $R_{s,t}$ such that for all $r \geq R_{s,t}$ then $ker(d_{r-1})^{s,t}$ is independent of $r$.

Moreover, $im(d_{r-1})$ consists of the image under $h$ of those $x \in D^{s-1,t}$ such that $g^{r-2}(x)$ is in the image of $f$, hence (since $im(f) = ker(g)$ by exactness of the exact couple) such that $g^{r-2}(x)$ is in the kernel of $g$, hence such that $x$ is in the kernel of $g^{r-1}$. If $r \gt R$ then by assumption $g^{r-1}|_{D^{s-1,t}} = 0$ and so then $im(d_{r-1}) = im(h)$.

(Beware this subtlety: while $g^{R_{s,t}}|_{D^{s-1,t}}$ vanishes by the convergence assumption, the expression $g^{R_{s,t}}|_{D^{s+r-1,t-r+1}}$ need not vanish yet. Only the higher power $g^{R_{s,t}+ R_{s+1,t+2}+2}|_{D^{s+r-1,t-r+1}}$ is again guaranteed to vanish. )

It follows that

\begin{aligned} E_\infty^{p,n-p} & = ker(d_R)/im(d_R) \\ & \simeq f^{-1}(im(g^{R-1}))/im(h) \\ & \simeq f^{-1}(im(g^{R-1}))/ker(f) \\ & \underoverset{\simeq}{f}{\longrightarrow} im(g^{R-1}) \cap im(f) \\ & \simeq im(g^{R-1}) \cap ker(g) \end{aligned}

where in last two steps we used once more the exactness of the exact couple.

(Notice that the above equation means in particular that the $E_\infty$-page is a sub-group of the image of the $E_1$-page under $f$.)

The last group above is that of elements $x \in G^n$ which map to zero in $D^{p-1,n-p+1}$ and where two such are identified if they agree in $D^{p,n-p}$, hence indeed

$E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n \,.$

## Examples

### Exact couple of a tower of (co)-fibrations

###### Definition

A filtered spectrum is a spectrum $X$ equipped with a sequence $X_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Spectra$ of spectra of the form

$\cdots \longrightarrow X_3 \stackrel{f_2}{\longrightarrow} X_2 \stackrel{f_1}{\longrightarrow} X_1 \stackrel{f_0}{\longrightarrow} X_0 = X \,.$
###### Remark

More generally a filtering on an object $X$ in (stable or not) homotopy theory is a $\mathbb{Z}$-graded sequence $X_\bullet$ such that $X$ is the homotopy colimit $X\simeq \underset{\longrightarrow}{\lim} X_\bullet$. But for the present purpose we stick with the simpler special case of def. .

###### Remark

There is no condition on the morphisms in def. . In particular, they are not required to be n-monomorphisms or n-epimorphisms for any $n$.

On the other hand, while they are also not explicitly required to have a presentation by cofibrations or fibrations, this follows automatically: by the existence of model structures for spectra, every filtering on a spectrum is equivalent to one in which all morphisms are represented by cofibrations or by fibrations.

This means that we may think of a filtration on a spectrum $X$ in the sense of def. as equivalently being a tower of fibrations over $X$.

The following remark unravels the structure encoded in a filtration on a spectrum, and motivates the concepts of exact couples and their spectral sequences from these.

###### Remark

Given a filtered spectrum as in def. , write $A_k$ for the homotopy cofiber of its $k$th stage, such as to obtain the diagram

$\array{ \cdots &\stackrel{}{\longrightarrow}& X_3 &\stackrel{f_2}{\longrightarrow}& X_2 &\stackrel{f_2}{\longrightarrow} & X_1 &\stackrel{f_1}{\longrightarrow}& X \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && A_3 && A_2 && A_1 && A_0 }$

where each stage

$\array{ X_{k+1} &\stackrel{f_k}{\longrightarrow}& X_k \\ && \downarrow^{\mathrlap{cofib(f_k)}} \\ && A_k }$

To break this down into invariants, apply the stable homotopy groups-functor. This yields a diagram of $\mathbb{Z}$-graded abelian groups of the form

$\array{ \cdots &\stackrel{}{\longrightarrow}& \pi_\bullet(X_3) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(X_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} & \pi_\bullet(X_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(X_0) \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0) } \,.$

Here each hook at stage $k$ extends to a long exact sequence of homotopy groups via connecting homomorphisms $\delta_\bullet^k$

$\cdots \to \pi_{\bullet+1}(A_k) \stackrel{\delta_{\bullet+1}^k}{\longrightarrow} \pi_\bullet(X_{k+1}) \stackrel{\pi_\bullet(f_k)}{\longrightarrow} \pi_\bullet(X_k) \stackrel{}{\longrightarrow} \pi_\bullet(A_k) \stackrel{\delta_\bullet^k}{\longrightarrow} \pi_{\bullet-1}(X_{k+1}) \to \cdots \,.$

If we understand the connecting homomorphism

$\delta_k \colon \pi_\bullet(A_k) \longrightarrow \pi_\bullet(X_{k+1})$

as a morphism of degree -1, then all this information fits into one diagram of the form

$\array{ \cdots &\stackrel{}{\longrightarrow}& \pi_\bullet(X_3) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(X_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} & \pi_\bullet(X_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(X_0) \\ && \downarrow &{}_{\mathllap{\delta_2}}\nwarrow & \downarrow &{}_{\mathllap{\delta_1}}\nwarrow & \downarrow &{}_{\mathllap{\delta_0}}\nwarrow & \downarrow \\ && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0) } \,,$

where each triangle is a rolled-up incarnation of a long exact sequence of homotopy groups (and in particular is not a commuting diagram!).

If we furthermore consider the bigraded abelian groups $\pi_\bullet(X_\bullet)$ and $\pi_\bullet(A_\bullet)$, then this information may further be rolled-up to a single diagram of the form

$\array{ \pi_\bullet(X_\bullet) & \stackrel{\pi_\bullet(f_\bullet)}{\longrightarrow} & \pi_\bullet(X_\bullet) \\ & {}_{\mathllap{\delta}}\nwarrow & \downarrow^{\mathrlap{\pi_\bullet(cofib(f_\bullet))}} \\ && \pi_\bullet(A_\bullet) }$

where the morphisms $\pi_\bullet(f_\bullet)$, $\pi_\bullet(cofib(f_\bullet))$ and $\delta$ have bi-degree $(0,-1)$, $(0,0)$ and $(-1,1)$, respectively.

Here it is convenient to shift the bigrading, equivalently, by setting

$\mathcal{D}^{s,t} \coloneqq \pi_{t-s}(X_s)$
$\mathcal{E}^{s,t} \coloneqq \pi_{t-s}(A_s) \,,$

because then $t$ counts the cycles of going around the triangles:

$\cdots \to \mathcal{D}^{s+1,t+1} \stackrel{\pi_{t-s}(f_s)}{\longrightarrow} \mathcal{D}^{s,t} \stackrel{\pi_{t-s}(cofib(f_s))}{\longrightarrow} \mathcal{E}^{s,t} \stackrel{\delta_s}{\longrightarrow} \mathcal{D}^{s+1,t} \to \cdots$

Data of this form is called an exact couple, def. below.

###### Definition

An unrolled exact couple (of Adams-type) is a diagram of abelian groups of the form

$\array{ \cdots &\stackrel{}{\longrightarrow}& \mathcal{D}^{3,\bullet} &\stackrel{i_2}{\longrightarrow}& \mathcal{D}^{2,\bullet} &\stackrel{i_1}{\longrightarrow} & \mathcal{D}^{1,\bullet} &\stackrel{i_0}{\longrightarrow}& \mathcal{D}^{0,\bullet} \\ && \downarrow^{\mathrlap{}} &{}_{\mathllap{k_2}}\nwarrow & {}^{\mathllap{j_2}}\downarrow &{}_{\mathllap{k_1}}\nwarrow & {}^{\mathllap{j_1}}\downarrow &{}_{\mathllap{k_0}}\nwarrow & {}_{\mathllap{j_0}}\downarrow \\ && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet} && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet} }$

such that each triangle is a rolled-up long exact sequence of abelian groups of the form

$\cdots \to \mathcal{D}^{s+1,t+1} \stackrel{i_s}{\longrightarrow} \mathcal{D}^{s,t} \stackrel{j_s}{\longrightarrow} \mathcal{E}^{s,t} \stackrel{k_s}{\longrightarrow} \mathcal{D}^{s+1,t} \to \cdots \,.$

The collection of this “un-rolled” data into a single diagram of abelian groups is called the corresponding exact couple.

###### Definition

An exact couple is a diagram (non-commuting) of abelian groups of the form

$\array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \,,$

such that this is exact sequence exact in each position, hence such that the kernel of every morphism is the image of the preceding one.

The concept of exact couple so far just collects the sequences of long exact sequences given by a filtration. Next we turn to extracting information from this sequence of sequences.

###### Remark

The sequence of long exact sequences in remark is inter-locking, in that every $\pi_{t-s}(X_s)$ appears twice:

$\array{ && & \searrow && \nearrow \\ && && \pi_{t-s-1}(X_{s+1}) \\ && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}} && && && \nearrow \\ && \pi_{t-s}(A_s) && \underset{def: \;\;d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) && \stackrel{def: \; d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2}) \\ & \nearrow && && && {}_{\mathllap{\delta_{t-s-1}^{s+1}}}\searrow && \nearrow_{\mathrlap{\pi_{t-s-2}(cofib(f_{s+2}))}} \\ && && && && \pi_{t-s-2}(X_{s+2}) \\ && && && & \nearrow && \searrow }$

This gives rise to the horizontal composites $d_1^{s,t}$, as show above, and by the fact that the diagonal sequences are long exact, these are differentials: $d_1^2 = 0$, hence give a chain complex:

$\array{ \cdots & \stackrel{}{\longrightarrow} && \pi_{t-s}(A_s) && \overset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) && \stackrel{d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2}) &&\longrightarrow & \cdots } \,.$

We read off from the interlocking long exact sequences what these differentials mean: an element $c \in \pi_{t-s}(A_s)$ lifts to an element $\hat c \in \pi_{t-s-1}(X_{s+2})$ precisely if $d_1 c = 0$:

$\array{ &\hat c \in & \pi_{t-s-1}(X_{s+2}) \\ && & \searrow^{\mathrlap{\pi_{t-s-1}(f_{s+1})}} \\ && && \pi_{t-s-1}(X_{s+1}) \\ && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}} \\ & c \in & \pi_{t-s}(A_s) && \underset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) }$

This means that the cochain cohomology of the complex $(\pi_{\bullet}(A_\bullet), d_1)$ produces elements of $\pi_\bullet(X_\bullet)$ and hence of $\pi_\bullet(X)$.

In order to organize this observation, notice that in terms of the exact couple of remark , the differential

$d_1^{s,t} \;\coloneqq \; \pi_{t-s-1}(cofib(f_{s+1})) \circ \delta_{t-s}^s$

is a component of the composite

$d \coloneqq j \circ k \,.$

Some terminology:

###### Definition

Given an exact couple, def. ,

$\array{ \mathcal{D}^{\bullet,\bullet} &\stackrel{i}{\longrightarrow}& \mathcal{D}^{\bullet,\bullet} \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E}^{\bullet,\bullet} }$

its page is the chain complex

$(E^{\bullet,\bullet}, d \coloneqq j \circ k) \,.$
###### Definition

Given an exact couple, def. , then the induced derived exact couple is the diagram

$\array{ \widetilde {\mathcal{D}} &\stackrel{\tilde i}{\longrightarrow}& \widetilde {\mathcal{D}} \\ & {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}} \\ && \widetilde{\mathcal{E}} }$

with

1. $\tilde{\mathcal{E}} \coloneqq ker(d)/im(d)$;

2. $\tilde {\mathcal{D}} \coloneqq im(i)$;

3. $\tilde i \coloneqq i|_{im(i)}$;

4. $\tilde j \coloneqq j \circ (im(i))^{-1}$;

5. $\tilde k \coloneqq k|_{ker(d)}$.

###### Proposition

A derived exact couple, def. , is again an exact couple, def. .

###### Definition

Given an exact couple, def. , then the induced spectral sequence, def. , is the sequence of pages, def. , of the induced sequence of derived exact couples, def. , prop. .

###### Example

Consider a filtered spectrum, def. ,

$\array{ \cdots &\stackrel{}{\longrightarrow}& X_3 &\stackrel{f_2}{\longrightarrow}& X_2 &\stackrel{f_2}{\longrightarrow} & X_1 &\stackrel{f_1}{\longrightarrow}& X \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && A_3 && A_2 && A_1 && A_0 }$

and its induced exact couple of stable homotopy groups, from remark

$\array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{k}}\nwarrow& \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \;\;\;\;\;\,\;\;\;\;\;\; \array{ \mathcal{D} &\stackrel{(-1,-1)}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{(1,0)}}\nwarrow& \downarrow^{\mathrlap{(0,0)}} \\ && \mathcal{E} }$

with bigrading as shown on the right. As we pass to derived exact couples, by def. , the bidegree of $i$ and $k$ is preserved, but that of $j$ increases by $(1,1)$ in each step, since

$deg(\tilde j) = deg( j \circ im(i)^{-1}) = deg(j) + (1,1) \,.$

Therefore the induced spectral sequence has differentials of the form

$d_r \;\colon\; \mathcal{E}_r^{s,t} \longrightarrow \mathcal{E}_r^{s+r, t+r-1} \,.$

The original article is

• W. S. Massey, Exact Couples in Algebraic Topology (Parts I and II), Annals of Mathematics, Second Series, Vol. 56, No. 2 (Sep., 1952), pp. 363-396 (pdf)

also

A class of examples leading to what later came to be known as the Atiyah-Hirzebruch spectral sequence is discussed in section XV.7 of

Textbook accounts include

Another review with an eye towards application to the Adams spectral sequence is in