What kind of spectral sequences come from double complexes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:56:01Z http://mathoverflow.net/feeds/question/110812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110812/what-kind-of-spectral-sequences-come-from-double-complexes What kind of spectral sequences come from double complexes? Hiro 2012-10-27T05:54:39Z 2012-10-29T18:27:44Z <p>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.</p> <p>My question is: When is a (homological or cohomological) spectral sequence coming from a double complex?</p> http://mathoverflow.net/questions/110812/what-kind-of-spectral-sequences-come-from-double-complexes/110820#110820 Answer by Johannes Ebert for What kind of spectral sequences come from double complexes? Johannes Ebert 2012-10-27T11:25:17Z 2012-10-29T18:27:44Z <p>There are two different ways to understand the question: </p> <ol> <li><p>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.</p></li> <li><p>For wich set of names ''$XY$''; the $XY$-spectral sequence can be derived from a double complex?</p></li> </ol> <p>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.</p> <p>Examples:</p> <ol> <li><p>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. </p></li> <li><p>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''.</p></li> <li><p>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.</p></li> <li><p>The Eilenberg-Moore spectral sequence comes from a double complex.</p></li> <li><p>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.</p></li> </ol> <p>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 $X$.</p>