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exact couple

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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 π’ž\mathcal{C}, an exact couple in π’ž\mathcal{C} is a cyclic long exact sequence of three morphisms among two objects of the form

β‹―βŸΆkEβ†’jDβŸΆΟ†Dβ†’kE⟢jβ‹―. \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

D βŸΆΟ† D jβ†– ↙ k E \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 \ref{ExactCoupleOfATower} below.

Spectral sequences from exact couples

Definition

A cohomology spectral sequence {E r p,q,d r}\{E_r^{p,q}, d_r\} is

  1. a sequence {E r β€’,β€’}\{E_r^{\bullet,\bullet}\} rβˆˆβ„€r \in \mathbb{Z}, rβ‰₯2r \geq 2 of bigraded abelian groups;

  2. a sequence of differentials {d r:E r β€’,β€’βŸΆE r β€’+r,β€’βˆ’r+1}\{d_r \colon E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}

such that

  • H r+1 β€’,β€’H_{r+1}^{\bullet,\bullet} is the cochain cohomology of d rd_r:, i.e. E r+1 β€’,β€’=H(E r β€’,β€’,d r)E_{r+1}^{\bullet, \bullet} = H(E_r^{\bullet,\bullet},d_r).

Given a β„€\mathbb{Z}-graded abelian group_ C β€’C^\bullet equipped with a decreasing filtration

C β€’βŠƒβ‹―βŠƒF sC β€’βŠƒF s+1C β€’βŠƒβ‹―βŠƒ0 C^\bullet \supset \cdots \supset F^s C^\bullet \supset F^{s+1} C^\bullet \supset \cdots \supset 0

such that

C β€’=βˆͺsF sC β€’and0=∩sF sC β€’ 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 β€’C^\bullet, denoted,

E 2 β€’,β€’β‡’C β€’ E_2^{\bullet,\bullet} \Rightarrow C^\bullet

if

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

    E ∞ s,t≔E ≫1 s,tE_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t};

  2. E ∞ β€’,β€’E_\infty^{\bullet,\bullet} is the associated graded of the filtered C β€’C^\bullet in that

    E ∞ s,t≃F sC s+t/F s+1C s+tE_\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 β€’,β€’}\{E_2^{\bullet,\bullet}\} is equipped with the structure of a bigraded object algebra;

  2. F β€’C β€’F^\bullet C^\bullet is equipped with the structure of a filtered graded algebra (F pC kβ‹…F qC lβŠ‚F p+qC k+lF^p C^k \cdot F^q C^l \subset F^{p+q} C^{k+l});

such that

  1. each d rd_{r} is a derivation with respect to the (induced) algebra structure on E r β€’,β€’{E_r^{\bullet,\bullet}}, graded of degree 1 with respect to total degree;

  2. the multiplication on E ∞ β€’,β€’E_\infty^{\bullet,\bullet} is compatible with that on C β€’C^\bullet.

Definition

(derived exact couples)

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

D ⟢g D fβ†– ↙ h E \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(f). im(h) = ker(f)\,,\;\;\; im(f) = ker(g)\,, \;\;\; im(g) = ker(f) \,.

Given an exact couple, then its derived exact couple is

im(g) ⟢g im(g) fβ†– ↙ h∘g βˆ’1 H(E,h∘f). \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 βˆ’1g^{-1} etc. for the operation of choosing a preimage under a given function gg. 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. 3,

D 1 ⟢g 1 D 1 f 1β†– ↙ h 1 E 1 \array{ D_1 && \stackrel{g_1}{\longrightarrow} && D_1 \\ & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}} \\ && E_1 }

its derived exact couple

D 2 ⟢g 2 D 2 f 2β†– ↙ h 2 E 2 \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

D r ⟢g r D r f rβ†– ↙ h r E r. \array{ D_r && \stackrel{g_r}{\longrightarrow} && D_r \\ & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}} \\ && E_r } \,.

If the abelian groups DD and EE are equipped with bigrading such that

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

then {E r β€’,β€’,d r}\{E_r^{\bullet,\bullet}, d_r\} with

d r ≔h r∘f r =h∘g βˆ’r+1∘f \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. 2.

(As before in prop. 1, the notation g βˆ’ng^{-n} with nβˆˆβ„•n \in \mathbb{N} denotes the function given by choosing, on representatives, a preimage under g n=gβˆ˜β‹―βˆ˜g∘g⏟ntimesg^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)(s,t) there exists R s,t≫1R_{s,t} \gg 1 such that for all rβ‰₯R s,tr \geq R_{s,t}

  1. g:D s+R,tβˆ’RβŸΆβ‰ƒD s+Rβˆ’1,tβˆ’Rβˆ’1g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1};

  2. g:D sβˆ’R+1,t+Rβˆ’2⟢0D sβˆ’R,t+Rβˆ’1g\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 ‒≔lim(β‹―β†’gD s,β€’βˆ’s⟢gD sβˆ’1,β€’βˆ’s+1β†’gβ‹―) 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 pG ‒≔ker(G β€’β†’D pβˆ’1,β€’βˆ’p+1). 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 ∞E_\infty-page:

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

ker(d rβˆ’1)≔ker(h∘g βˆ’r+2∘f) ker(d_{r-1}) \coloneqq ker(h \circ g^{-r+2} \circ f)

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

ker(d rβˆ’1) s,t≃f βˆ’1(g rβˆ’1(D s+rβˆ’1,tβˆ’r+1)). 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)(s,t) an R s,tR_{s,t} such that for all rβ‰₯R s,tr \geq R_{s,t} then ker(d rβˆ’1) s,tker(d_{r-1})^{s,t} is independent of rr.

Moreover, im(d rβˆ’1)im(d_{r-1}) consists of the image under hh of those x∈D sβˆ’1,tx \in D^{s-1,t} such that g rβˆ’2(x)g^{r-2}(x) is in the image of ff, hence (since im(f)=ker(g)im(f) = ker(g) by exactness of the exact couple) such that g rβˆ’2(x)g^{r-2}(x) is in the kernel of gg, hence such that xx is in the kernel of g rβˆ’1g^{r-1}. If r>Rr \gt R then by assumption g rβˆ’1| D sβˆ’1,t=0g^{r-1}|_{D^{s-1,t}} = 0 and so then im(d rβˆ’1)=im(h)im(d_{r-1}) = im(h).

(Beware this subtlety: while g R s,t| D sβˆ’1,tg^{R_{s,t}}|_{D^{s-1,t}} vanishes by the convergence assumption, the expression g R s,t| D s+rβˆ’1,tβˆ’r+1g^{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+rβˆ’1,tβˆ’r+1g^{R_{s,t}+ R_{s+1,t+2}+2}|_{D^{s+r-1,t-r+1}} is again guaranteed to vanish. )

It follows that

E ∞ p,nβˆ’p =ker(d R)/im(d R) ≃f βˆ’1(im(g Rβˆ’1))/im(h) ≃f βˆ’1(im(g Rβˆ’1))/ker(f) βŸΆβ‰ƒfim(g Rβˆ’1)∩im(f) ≃im(g Rβˆ’1)∩ker(g) \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 ∞E_\infty-page is a sub-group of the image of the E 1E_1-page under ff.)

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

E ∞ p,nβˆ’p≃F pG n/F p+1G n. E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n \,.

Examples

Exact couple of a tower of (co)-fibrations

…spectral sequence of a tower of fibrations…

Definition

A filtered spectrum is a spectrum XX equipped with a sequence X β€’:(β„•,>)⟢SpectraX_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Spectra of spectra of the form

β‹―βŸΆX 3⟢f 2X 2⟢f 1X 1⟢f 0X 0=X. \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 XX in (stable or not) homotopy theory is a β„€\mathbb{Z}-graded sequence X β€’X_\bullet such that XX is the homotopy colimit X≃lim⟢X β€’X\simeq \underset{\longrightarrow}{\lim} X_\bullet. But for the present purpose we stick with the simpler special case of def. 4.

Remark

There is no condition on the morphisms in def. 4. In particular, they are not required to be n-monomorphisms or n-epimorphisms for any nn.

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 XX in the sense of def. 4 as equivalently being a tower of fibrations over XX.

The following remark 4 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. 4, write A kA_k for the homotopy cofiber of its kkth stage, such as to obtain the diagram

β‹― ⟢ X 3 ⟢f 2 X 2 ⟢f 2 X 1 ⟢f 1 X ↓ ↓ ↓ ↓ A 3 A 2 A 1 A 0 \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

X k+1 ⟢f k X k ↓ cofib(f k) A k \array{ X_{k+1} &\stackrel{f_k}{\longrightarrow}& X_k \\ && \downarrow^{\mathrlap{cofib(f_k)}} \\ && A_k }

is a homotopy fiber sequence.

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

β‹― ⟢ Ο€ β€’(X 3) βŸΆΟ€ β€’(f 2) Ο€ β€’(X 2) βŸΆΟ€ β€’(f 2) Ο€ β€’(X 1) βŸΆΟ€ β€’(f 1) Ο€ β€’(X 0) ↓ ↓ ↓ ↓ Ο€ β€’(A 3) Ο€ β€’(A 2) Ο€ β€’(A 1) Ο€ β€’(A 0). \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 kk extends to a long exact sequence of homotopy groups via connecting homomorphisms Ξ΄ β€’ k\delta_\bullet^k

β‹―β†’Ο€ β€’+1(A k)⟢δ β€’+1 kΟ€ β€’(X k+1)βŸΆΟ€ β€’(f k)Ο€ β€’(X k)βŸΆΟ€ β€’(A k)⟢δ β€’ kΟ€ β€’βˆ’1(X k+1)β†’β‹―. \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

Ξ΄ k:Ο€ β€’(A k)βŸΆΟ€ β€’(X k+1) \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

β‹― ⟢ Ο€ β€’(X 3) βŸΆΟ€ β€’(f 2) Ο€ β€’(X 2) βŸΆΟ€ β€’(f 2) Ο€ β€’(X 1) βŸΆΟ€ β€’(f 1) Ο€ β€’(X 0) ↓ Ξ΄ 2β†– ↓ Ξ΄ 1β†– ↓ Ξ΄ 0β†– ↓ Ο€ β€’(A 3) Ο€ β€’(A 2) Ο€ β€’(A 1) Ο€ β€’(A 0), \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 Ο€ β€’(X β€’)\pi_\bullet(X_\bullet) and Ο€ β€’(A β€’)\pi_\bullet(A_\bullet), then this information may further be rolled-up to a single diagram of the form

Ο€ β€’(X β€’) βŸΆΟ€ β€’(f β€’) Ο€ β€’(X β€’) Ξ΄β†– ↓ Ο€ β€’(cofib(f β€’)) Ο€ β€’(A β€’) \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 Ο€ β€’(f β€’)\pi_\bullet(f_\bullet), Ο€ β€’(cofib(f β€’))\pi_\bullet(cofib(f_\bullet)) and Ξ΄\delta have bi-degree (0,βˆ’1)(0,-1), (0,0)(0,0) and (βˆ’1,1)(-1,1), respectively.

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

π’Ÿ s,t≔π tβˆ’s(X s) \mathcal{D}^{s,t} \coloneqq \pi_{t-s}(X_s)
β„° s,t≔π tβˆ’s(A s), \mathcal{E}^{s,t} \coloneqq \pi_{t-s}(A_s) \,,

because then tt counts the cycles of going around the triangles:

β‹―β†’π’Ÿ s+1,t+1βŸΆΟ€ tβˆ’s(f s)π’Ÿ s,tβŸΆΟ€ tβˆ’s(cofib(f s))β„° s,t⟢δ sπ’Ÿ s+1,tβ†’β‹― \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. 6 below.

Definition

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

β‹― ⟢ π’Ÿ 3,β€’ ⟢i 2 π’Ÿ 2,β€’ ⟢i 1 π’Ÿ 1,β€’ ⟢i 0 π’Ÿ 0,β€’ ↓ k 2β†– j 2↓ k 1β†– j 1↓ k 0β†– j 0↓ β„° 3,β€’ β„° 2,β€’ β„° 1,β€’ β„° 0,β€’ \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

β‹―β†’π’Ÿ s+1,t+1⟢i sπ’Ÿ s,t⟢j sβ„° s,t⟢k sπ’Ÿ s+1,tβ†’β‹―. \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 β€œun-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

π’Ÿ ⟢i π’Ÿ kβ†– ↓ j β„°, \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 4 is inter-locking, in that every Ο€ tβˆ’s(X s)\pi_{t-s}(X_s) appears twice:

β†˜ β†— Ο€ tβˆ’sβˆ’1(X s+1) Ξ΄ tβˆ’s sβ†— β†˜ Ο€ tβˆ’sβˆ’1(cofib(f s+1)) β†— Ο€ tβˆ’s(A s) ⟢def:d 1 s,t Ο€ tβˆ’sβˆ’1(A s+1) ⟢def:d 1 s+1,t Ο€ tβˆ’sβˆ’2(A s+2) β†— Ξ΄ tβˆ’sβˆ’1 s+1β†˜ β†— Ο€ tβˆ’sβˆ’2(cofib(f s+2)) Ο€ tβˆ’sβˆ’2(X s+2) β†— β†˜ \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,td_1^{s,t}, as show above, and by the fact that the diagonal sequences are long exact, these are differentials: d 1 2=0d_1^2 = 0, hence give a chain complex:

β‹― ⟢ Ο€ tβˆ’s(A s) ⟢d 1 s,t Ο€ tβˆ’sβˆ’1(A s+1) ⟢d 1 s+1,t Ο€ tβˆ’sβˆ’2(A s+2) ⟢ β‹―. \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 cβˆˆΟ€ tβˆ’s(A s)c \in \pi_{t-s}(A_s) lifts to an element c^βˆˆΟ€ tβˆ’sβˆ’1(X s+2)\hat c \in \pi_{t-s-1}(X_{s+2}) precisely if d 1c=0d_1 c = 0:

c^∈ Ο€ tβˆ’sβˆ’1(X s+2) β†˜ Ο€ tβˆ’sβˆ’1(f s+1) Ο€ tβˆ’sβˆ’1(X s+1) Ξ΄ tβˆ’s sβ†— β†˜ Ο€ tβˆ’sβˆ’1(cofib(f s+1)) c∈ Ο€ tβˆ’s(A s) ⟢d 1 s,t Ο€ tβˆ’sβˆ’1(A s+1) \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 (Ο€ β€’(A β€’),d 1)(\pi_{\bullet}(A_\bullet), d_1) produces elements of Ο€ β€’(X β€’)\pi_\bullet(X_\bullet) and hence of Ο€ β€’(X)\pi_\bullet(X).

In order to organize this observation, notice that in terms of the exact couple of remark 4, the differential

d 1 s,t≔π tβˆ’sβˆ’1(cofib(f s+1))∘δ tβˆ’s s 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≔j∘k. d \coloneqq j \circ k \,.

Some terminology:

Definition

Given an exact couple, def. 6,

π’Ÿ β€’,β€’ ⟢i π’Ÿ β€’,β€’ kβ†– ↓ j β„° β€’,β€’ \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 β€’,β€’,d≔j∘k). (E^{\bullet,\bullet}, d \coloneqq j \circ k) \,.
Definition

Given an exact couple, def. 6, then the induced derived exact couple is the diagram

π’ŸΛœ ⟢i˜ π’ŸΛœ kΛœβ†– ↓ j˜ β„°Λœ \array{ \widetilde {\mathcal{D}} &\stackrel{\tilde i}{\longrightarrow}& \widetilde {\mathcal{D}} \\ & {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}} \\ && \widetilde{\mathcal{E}} }

with

  1. β„°Λœβ‰”ker(d)/im(d)\tilde{\mathcal{E}} \coloneqq ker(d)/im(d);

  2. π’ŸΛœβ‰”im(i)\tilde {\mathcal{D}} \coloneqq im(i);

  3. iΛœβ‰”i| im(i)\tilde i \coloneqq i|_{im(i)};

  4. jΛœβ‰”j∘(im(i)) βˆ’1\tilde j \coloneqq j \circ (im(i))^{-1};

  5. kΛœβ‰”k| ker(d)\tilde k \coloneqq k|_{ker(d)}.

Proposition

A derived exact couple, def. 8, is again an exact couple, def. 6.

Definition

Given an exact couple, def. 6, then the induced spectral sequence, def. \ref{SpectralSequence}, is the sequence of pages, def. 7, of the induced sequence of derived exact couples, def. 8, prop. 2.

Example

Consider a filtered spectrum, def. 4,

β‹― ⟢ X 3 ⟢f 2 X 2 ⟢f 2 X 1 ⟢f 1 X ↓ ↓ ↓ ↓ A 3 A 2 A 1 A 0 \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 4

π’Ÿ ⟢i π’Ÿ kβ†– ↓ j β„°π’Ÿ ⟢(βˆ’1,βˆ’1) π’Ÿ (1,0)β†– ↓ (0,0) β„° \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. 8, the bidegree of ii and kk is preserved, but that of jj increases by (1,1)(1,1) in each step, since

deg(j˜)=deg(j∘im(i) βˆ’1)=deg(j)+(1,1). 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:β„° r s,tβŸΆβ„° r s+r,t+rβˆ’1. d_r \;\colon\; \mathcal{E}_r^{s,t} \longrightarrow \mathcal{E}_r^{s+r, t+r-1} \,.

References

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

Revised on September 1, 2016 16:04:10 by Jonas Frey (128.237.172.84)