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Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.

My question is: When is a (homological or cohomological) spectral sequence coming from a double complex?

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There are two different ways to understand the question:

  1. If I see an abstract spectral seqeunce, is there a double complex such that its spectral sequence is isomorphic to the given spectral sequence? I do not have an answer to that question and, to be honest, do not believe it is an interesting question.

  2. For wich set of names ''$XY$''; the $XY$-spectral sequence can be derived from a double complex?

The answer is that, as a general rule (it might have exceptions), all $XY$-spectral sequences whose $E_2$-terms and $E_{\infty}$ terms are purely homological can be derived from filtered complexes; and most of them in fact from double complexes.

Examples:

  1. The spectral sequence of a simplicial space (Segal; ''Classfying spaces and spectral sequences'') can be reformulated using a double complex (a simplicial space $X_{\bullet}$ gives rise to a simplicial chain complex $C_{\ast} X_{\bullet}$ and thus a double complex.

  2. The Serre spectral sequence is a special case of the above; a direct construction using a double complex was given by A. Dress, ''Zur Spectralsequenz von Faserungen''.

  3. Special cases of 2. include the Lyndon-Hochschild-Serre spectral sequ. for group extensions; special cases of 1. include the Bousfield-Kan spectral sequ. of a homotopy colimit and some others.

  4. The Eilenberg-Moore spectral sequence comes from a double complex.

  5. Purely algebraic versions: Grothendieck-spectral sequence. Probably the spectral sequence of a Lie algebra extension fits into here. The Van Est spectral sequence for Lie algebra cohomology also comes from a double complex.

The Bockstein spectral sequence is a purely homological construction, it can be derived from a filtered complex; but it does not seem to stem from a double complex. Other counterexamples are the typical spectral sequence of stable homotopy theory (Atiyah-Hirzebruch, Adams spectral sequence): they cannot be derived from filtered complexes. In fact, if $E$ is a generalized homology theory, you cannot write $E_{\ast} (X)$ of a space in a sensible way as the homology of a chain complex functorially associated with $X$.

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Thanks a lot! The couterexamples are interesting. –  Hiro Nov 2 '12 at 16:59
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