You could have invented spectral sequences

Discussion in 'Math Research' started by tchow, Jun 21, 2005.

  1. tchow

    tchow Guest

    Eight years ago, I posted a link here to an expository article of mine
    on spectral sequences. The article has been available on my website,
    but I have always been conscious of its numerous shortcomings. I have
    now finally gotten around to cleaning it up; the revised version may be
    found at:

    As before, the goal of the article is to explain to the beginner not only
    what spectral sequences are, but how one might have discovered them in the
    first place. I have made numerous changes to the original version, each
    small, but collectively yielding what I feel is a much better exposition.

    By the way, thanks to Nath Rao for pointing me to the algebraic topology
    mailing list in response to my recent question on where the word
    "spectral" came from. I got a similar response from writing directly
    to John McCleary. This has been incorporated as a remark in the paper.
    tchow, Jun 21, 2005
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  2. tchow

    Per Vognsen Guest

    Very nice!

    Have you thought of adding some motivations from topology or are you
    determined to remain "application-agnostic"? It seems like a couple of
    remarks on how graded and filtered complexes arise naturally in
    topology might be very helpful to someone trying to understand this for
    the first time.

    For instance, gradings of chain complexes typically come from a
    disjoint sum decomposition of a space into its connected components.
    From this point of view it's both obvious why the homology splits into
    the homology of the graded pieces and also why it's not very deep or
    interesting to someone doing nontrivial calculations.

    Filtrations of chain complexes naturally arise from filtrations of a
    space by closed subspaces. A particular well-known example is a pair
    (X,A) which gives a two-term filtration. Here we know that homology
    calculations can be approached in a divide-and-conquer fashion using
    the long exact sequence for the pair. More generally we have an
    arbitrary filtration and would like a similar divide-and-conquer
    calculational approach. One way is to break a filtration A_1 sub A_2
    sub ... sub A_n = X into the series of two-term filtrations A_1 sub
    A_2, A_2 sub A_3, ..., A_(n-1) sub A_n = X and then proceed to
    calculate the homology from bottom to top using the long exact sequence
    for each of the pairs. This approach is great when there are only
    "short-range" interactions in the filtration--for instance, if X is a
    CW-complex and we are dealing with a filtration by its skeleta.
    Generally however, we have to consider "long-range" interactions
    between the pieces of the filtration and this provides motivation for
    the more heavy machinery you discuss in your article.

    Hopefully I'm not completely crazy. :)

    Per Vognsen, Jun 22, 2005
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