# You could have invented spectral sequences

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

1. ### tchowGuest

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:

http://alum.mit.edu/www/tchow/spectral02.pdf

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

2. ### Per VognsenGuest

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

Per Vognsen, Jun 22, 2005