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Exact couple

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In mathematics, an exact couple, due to 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.

Contents

For the definition of an exact sequence and the construction of a spectral sequence from it (which is immediate), see spectral sequence#Exact couples. 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 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 ⊂ F p C .

From the filtration one can form the associated graded complex:

gr ⁡ C = ⨁ − ∞ ∞ 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 = ( 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 → F p − 1 C → F p C → ( gr ⁡ C ) p → 0

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

⋯ → H n ( F p − 1 C ) → i H n ( F p C ) → j H n ( gr ⁡ ( C ) p ) → k H n − 1 ( F p − 1 C ) → ⋯

With the notation D p , q = H p + q ( F p C ) , E p , q 1 = H p + q ( gr ⁡ ( C ) p ) , the above reads:

⋯ → D p − 1 , q + 1 → i D p , q → j E p , q 1 → k D p − 1 , q → ⋯ ,

which is precisely an exact couple and E 1 is a complex with the differential d = j ∘ 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 → k D p − 1 , q r → r j 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).

Sketch of proof: Remembering d = j ∘ k , it is easy to see:

Z r = k − 1 ( 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 → F p C / F p − 1 C . Now, if [ x ¯ ] ∈ Z p , q r − 1 ⊂ E p , q 1 , then k ( [ x ¯ ] ) = i r − 1 ( [ y ] ) for some [ y ] ∈ D p − r , q + r − 1 = H p + q − 1 ( F p C ) . On the other hand, remembering k is a connecting homomorphism, k ( [ 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 ′ ∈ F p − 1 C . Hence, [ x ¯ ] ∈ Z p r ⇔ x ∈ A p r modulo F p − 1 C , yielding Z p r ≃ ( 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 ) → H p + q ( F p + r − 1 C ) ) is represented by a cycle x such that x ∈ d ( F p + r − 1 C ) . Hence, since j is induced by ⋅ ¯ , B p r − 1 = j ( ker ⁡ i r − 1 ) ≃ ( d ( A p + r − 1 r − 1 ) + F p − 1 C ) / F p − 1 C .

We conclude: since A p r ∩ F p − 1 C = A p − 1 r − 1 ,

E p , ∗ r = Z p r − 1 B p r − 1 ≃ A p r + F p − 1 C d ( A p + r − 1 r − 1 ) + F p − 1 C ≃ A p r d ( A p + r − 1 r − 1 ) + A p − 1 r − 1 . ◻

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, 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 = ⨁ i ≥ p K i , ∗ , for each with fixed p, we have the exact sequence of cochain complexes:

0 → G p + 1 → G p → K p , ∗ → 0.

Taking cohomology of it gives rise to an exact couple:

⋯ → D p , q → j E 1 p , q → k ⋯

where we used the notation 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 → E → 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

Exact couple Wikipedia
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