Spectral sequence

A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary Abelian category (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).

All known spectral sequences can be obtained from exact couples. An exact couple $ ( D ^ {1} , E ^ {1} , i ^ {1} , j ^ {1} , k ^ {1} ) $ is defined as an exact diagram of the form

$$ \begin{array}{lcccl} D ^ {1} &{} & \rightarrow ^ { {i ^ {1}} } &{} &D ^ {1} \\ {} &{} _ {k ^ {1} } &{} &{} _ {j ^ {1} } &{} \\ {} &{} &E ^ {1} &{} &{} \\ \end{array} $$

The homomorphism $ d ^ {1} = j ^ {1} k ^ {1} $ is a differential in $ E ^ {1} $. From any exact couple one can construct the derived exact couple $ ( D ^ {2} , E ^ {2} , i ^ {2} , j ^ {2} , k ^ {2} ) $, for which $ D ^ {2} = \mathop{\rm Im} i ^ {1} $ and $ E ^ {2} = H( E ^ {1} , d ^ {1} ) $. By iterating this construction one obtains the spectral sequence $ E = \{ E ^ {n} , d ^ {n} \} $.

1) The Leray spectral sequence. A filtered chain complex of modules $ ( \{ K ^ {p} \} , d) $ determines an exact couple of bigraded modules $ D _ {p,q} ^ {1} = H _ {p+q} ( K ^ {p} ) $, $ E _ {p,q} ^ {1} = H _ {p+q} ( K ^ {p} / K ^ {p-1} ) $. In the associated spectral sequence, the bidegree of the differential $ d ^ {r} $ is equal to $ (- r, r- 1) $, and

$$ E _ {p,q} ^ {r} = \frac{ \mathop{\rm Ker} ( d _ {p,q} ^ {r-1} : E _ {p,q} ^ {r-1} \rightarrow E _ {p-r+ 1,q+ r- 2} ^ {r-1} ) }{ \mathop{\rm Im} ( d _ {p+r- 1,q- r+ 2} ^ {r-1} : E _ {p+r- 1,q- r+ 2} ^ {r-1} \rightarrow E _ {p,q} ^ {r-1} ) } \simeq $$

$$ \simeq \frac{ \mathop{\rm Im} ( H _ {p+q} ( K ^ {p} / K ^ {p-r} ) \rightarrow H _ {p+q} ( K ^ {p} / K ^ {p-1} )) }{ \mathop{\rm Im} ( \partial : H _ {p+q+ 1} ( K ^ {p+r- 1} / K ^ {p} ) \rightarrow H _ {p+q} ( K ^ {p} / K ^ {p-1} )) } . $$

The modules $ F _ {p,q} = \mathop{\rm Im} ( H _ {p+q} ( K ^ {p} ) \rightarrow H _ {p+q} ( K)) $ form a filtration of $ H _ {*} ( K) $. The bigraded module

$$ E _ {p,q} ^ \infty = F _ {p,q} / F _ {p-1,q+ 1} \simeq $$

$$ \simeq \frac{ \mathop{\rm Im} ( H _ {p+q} ( K ^ {p} ) \rightarrow H _ {p+q} ( K ^ {p} / K ^ {p-1} )) }{ \mathop{\rm Im} ( \partial : H _ {p+q+ 1} ( K / K ^ {p} ) \rightarrow H _ {p+q} ( K ^ {p} / K ^ {p-1} )) } $$

is called the associated graded module of $ H _ {*} ( K) $. The filtration $ \{ K ^ {p} \} $ is called regular if $ K ^ {p} = 0 $ when $ p< 0 $, $ E _ {p,q} ^ {1} = 0 $ when $ q< 0 $ and $ K = \cup K ^ {p} $. For a regular filtration, $ E _ {p,q} ^ {r} = 0 $ when $ p< 0 $ or $ q< 0 $; such a spectral sequence is called a first-quadrant spectral sequence. Moreover, $ E _ {p,q} ^ {r} \simeq E _ {p,q} ^ {r+1} \simeq E _ {p,q} ^ \infty $ when $ r > \max ( p, q+ 1) $. In this case one says that the spectral sequence converges to $ H _ {*} ( K) $, and writes $ E _ {p,q} ^ {r} \Rightarrow H _ {p+q} ( K) $.

2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a CW-complex $ X $ by its skeletons gives the collapsing spectral sequence $ E _ {p,q} ^ {r} \Rightarrow H _ {p+q} ( X) $, for which $ E _ {p,q} ^ {r} = E _ {p,q} ^ \infty = 0 $ when $ q \neq 0 $ and $ E _ {n,0} ^ {2} = E _ {n,0} ^ \infty = H _ {n} ( X) $. The Leray–Serre spectral sequence is obtained from the filtration of the total space $ E $ of the Serre fibration $ F \rightarrow ^ {i} E \rightarrow ^ {p} B $ by the pre-images $ p ^ {-1} ( B ^ {n} ) $ of the skeletons $ B ^ {n} $ of the base $ B $. If the fibre $ F $ and base $ B $ are path-connected, then for every coefficient group $ G $ this gives the spectral sequence $ E _ {p,q} ^ {r} \Rightarrow H _ {p+q} ( E, G) $ with differentials $ d ^ {r} $ of bidegree $ ( - r, r- 1) $ for which

$$ E _ {p,q} ^ {1} \simeq C _ {p} ( B) \otimes H _ {q} ( F; G) \ \textrm{ and } \ E _ {p,q} ^ {2} \simeq H _ {p} ( B; {\mathcal H} _ {q} ( F; G)), $$

where $ {\mathcal H} _ {q} ( F; G) $ is a system of local coefficients over $ B $ consisting of the groups $ H _ {q} ( F; G) $. The homomorphism $ i _ {*} : H _ {n} ( F; G) \rightarrow H _ {n} ( E; G) $ coincides with the composite

$$ H _ {n} ( F; G) = E _ {0,n} ^ {2} \rightarrow E _ {0,n} ^ {r} = E _ {0,n} ^ \infty = F _ {0,n} \subset H _ {n} ( F; G), $$

and the homomorphism $ p _ {*} : H _ {n} ( E; G) \rightarrow H _ {n} ( B; G) $ coincides with the composite

$$ H _ {n} ( E; G) = F _ {n,0} \rightarrow E _ {n,0} ^ \infty = \ E _ {n,0} ^ {r} \subset \ E _ {n,0} ^ {2} = H _ {n} ( B; G), $$

where $ r $ is sufficiently large. The differential $ d _ {n,0} ^ {n} $ of the spectral sequence coincides with the transgression $ \tau : H _ {n} ( B; G) \rightarrow H _ {n-1} ( F; G) $.

This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence $ E _ {r} ^ {p,q} \Rightarrow H ^ {p+q} ( E; G) $, with differentials $ d _ {r} $ of bidegree $ ( r, - r+ 1) $, for which $ E _ {2} ^ {p,q} \simeq H ^ {p} ( B; {\mathcal H} _ {q} ( F; G)) $. If $ G $ is a ring, then every term $ E _ {r} $ is a bigraded ring, $ d _ {r} $ is differentiation in $ E _ {r} $, and the multiplication in $ E _ {r+1} $ is induced by that in $ E _ {r} $. If $ G $ is a field and the base $ B $ is simply connected, then $ E _ {2} ^ {**} \simeq H ^ {*} ( B; G) \otimes H ^ {*} ( F; G) $.

3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor $ h _ {*} $ ($ h ^ {*} $) to the same filtration of the space $ E $. In its cohomological version, $ E _ {r} ^ {p,q} \Rightarrow h ^ {p+q} ( E) $, $ E _ {2} ^ {p,q} = H ^ {p} ( B; h ^ {q} ( F )) $. In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration $ \mathop{\rm id} : X \rightarrow X $ is in general non-collapsing.

4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations

$$ \begin{array}{ccc} E & \rightarrow & X \\ \downarrow &{} &\downarrow \\ Y & \rightarrow & B \\ \end{array} $$

In its cohomological version,

$$ E _ {r} \Rightarrow H ^ {*} ( E; R), $$

$$ E _ {2} ^ {p,q} \simeq \mathop{\rm Tor} _ {H ^ {*} ( B;R) } ^ {p,q} ( H ^ {*} ( X; R); H ^ {*} ( Y; R)). $$

If $ R $ is a field and the square consists of $ H $-spaces and $ H $-mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.

5) The Adams spectral sequence $ E _ {r} ^ {s,t} $ is defined for every prime $ p\geq 2 $ and all spaces $ X $ and $ Y $ (satisfying certain finiteness conditions). One has

$$ E _ {2} ^ {s,t} \simeq \mathop{\rm Ext} _ {A _ {p} } ^ {s,t} ( H ^ {*} ( X; \mathbf Z _ {p} ); H ^ {*} ( Y ; \mathbf Z _ {p} )), $$

where $ A _ {p} $ is the Steenrod algebra modulo $ p $. The bidegree of $ d _ {r} $ is equal to $ ( r, r- 1) $. This spectral sequence converges in the sense that, when $ r> s $, there is a monomorphism $ E _ {r+1} ^ {s,t} \rightarrow E _ {r} ^ {s,t} $, and so the group $ E _ \infty ^ {s,t} = \cap _ {r>s} E _ {r} ^ {s,t} $ is defined. There is a decreasing filtration $ \{ F ^ { s } \} $ of the group $ \{ Y, X \} $ of stable homotopy classes of mappings $ Y \rightarrow X $ such that

$$ F ^ { s } \{ S ^ {t-s} YX \} / F ^ { s+ 1 } \{ S ^ {t-s} Y, X \} \simeq E _ \infty ^ {s,t} , $$

and $ F ^ { \infty } = \cap _ {s\geq } 0 F ^ { s } $ consists of all elements of $ \{ Y, X \} $ of finite order prime with $ p $. When $ X= Y= S ^ {0} $, this spectral sequence enables one "in principle" to calculate the $ p $-components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary generalized cohomology theories. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.

References

[1]	R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[2]	D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[3]	J.-P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. , 54 (1951) pp. 425–505
[4]	S. MacLane, "Homology" , Springer (1963)
[5]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[6]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[7]	S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[8]	R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[9]	S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theory" Math. USSR Izv. , 31 (1967) pp. 827–913 Izv. Akad. Nauk. SSSR Ser. Mat. , 31 (1967) pp. 855–951
[10]	J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974)
[11]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[12]	L. Smith, "Lectures on the Eilenberg–Moore spectral sequence" , Lect. notes in math. , 134 , Springer (1970)
[13]	D.C. Ravenel, "A novices guide to the Adams–Novikov spectral sequence" , Geometric Applications of Homotopy Theory , 2 , Springer (1978) pp. 404–475

Comments

Let $ ( E ^ {n} , d ^ {n} ) $, $ n = 2, 3 \dots $ be a spectral sequence, so that $ E ^ {n+1} $ is the homology of $ ( E ^ {n} , d ^ {n} ) $. A spectral sequence defines a series of modules of the initial term $ E ^ {2} $, defined inductively as follows:

$$ 0 = B ^ {1} \subset B ^ {2} \subset B ^ {3} \subset \dots $$

$$ \dots \subset C ^ {3} \subset C ^ {2} \subset C ^ {1} = E ^ {2} , $$

with $ E ^ {r+1} = C ^ {r} / B ^ {r} $, and $ C ^ {r+1} /B ^ {r} $ is the kernel of $ d ^ {r} : E ^ {r} \rightarrow E ^ {r} $, while $ B ^ {r+1} /B ^ {r} $ is the image of $ d ^ {r} $. One now defines the infinity terms:

$$ C ^ \infty = \cap _ { n } C ^ {n} ,\ B ^ \infty = \ \cup _ { n } B ^ {n} ,\ E ^ \infty = C ^ \infty / B ^ \infty . $$

The terms $ E ^ {r} $ are thought of as successive approximations of $ E ^ \infty $. If $ ( E ^ {n} , d ^ {n} ) $ is a spectral sequence of bigraded modules $ E ^ {n} = \oplus E _ {p,q} ^ {n} $, $ d ^ {r} : E _ {p,q} ^ {r} \rightarrow E _ {p-r,q+ r- 1} ^ {r} $, all the $ B ^ {i} $, $ C ^ {i} $, $ B ^ \infty $, $ C ^ \infty $, $ E ^ \infty $ also carry corresponding natural bigraded structures.

Sometimes there is an initial term $ E ^ {1} $, and then the same construction is carried out with $ E ^ {1} $ instead of $ E ^ {2} $.

For a first-quadrant spectral sequence, i.e. $ E _ {p,q} ^ {2} = 0 $ for $ p< 0 $ or $ q< 0 $, for given $ p, q $ and large enough $ r $ one has that in

$$ E _ {p+r,q- r+ 1} ^ {r} \rightarrow ^ { {d ^ {r}} } E _ {p,q} ^ {r} \rightarrow ^ { {d ^ {r}} } E _ {p-r,q+ r- 1} ^ {r} $$

both the outside modules are zero, so that $ E _ {p,q} ^ {r} = E _ {p,q} ^ {r+1} = E _ {p,q} ^ \infty $ for $ r $ large enough.

For a first-quadrant spectral sequence one also always has that $ E _ {p,0} ^ {r+1} $ is a submodule of $ E _ {p,0} ^ {r} $, and $ E _ {0,q} ^ {r+1} $ is a quotient of $ E _ {0,q} ^ {r} $, giving rise to sequences of monomorphisms and epimorphisms:

$$ E _ {p,0} ^ \infty = E _ {p,0} ^ {p+1} \rightarrow \dots \rightarrow E _ {p,0} ^ {3} \rightarrow E _ {p,0} ^ {2} , $$

$$ E _ {0,q} ^ {2} \rightarrow E _ {0,q} ^ {3} \rightarrow \dots \rightarrow E _ {0,q} ^ {q+2} = E _ {0,q} ^ \infty , $$

which are known as the edge homomorphisms.

Let $ ( A _ {p} ) $ be a filtration of a module $ A $ by submodules

$$ \dots \subset A _ {p-1} \subset A _ {p} \subset A _ {p+1} \subset \dots $$

with associated graded module $ \mathop{\rm Gr} ( A) $:

$$ \mathop{\rm Gr} ( A) = \oplus _ { p } A _ {p} / A _ {p-1} . $$

A spectral sequence $ ( E _ {p} ^ {r} , d ^ {r} ) $ is said to converge to a graded module $ H $, in symbols

$$ E _ {p} ^ {r} \Rightarrow H , $$

if there is a filtration $ F _ {p} H $ of $ H $ such that

$$ \tag{* } E _ {p} ^ \infty \simeq F _ {p} H /F _ {p+1} H . $$

In the usual cases the $ E _ {p} ^ {r} $ and $ H $ are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.