An essay on spectral sequences

This is primarily an essay on the Serre spectral sequence for the homology/cohomology of a fibration. But we view the Serre spectral sequence as a special case of the spectral sequence of a filtered space or filtered chain complex, which has many other important cases: the spectral sequence(s) of a double complex (Grothendieck’s composite functor spectral sequence in homological algebra is obtained this way); the Atiyah-Hirzebruch spectral sequence for generalized homology theories; the Rothenberg-Steenrod spectral sequence for the homology of classifying spaces; and more. The goal is to explain the ideas behind the spectral sequence and to illustrate how it is used; technical details are largely omitted, and occasionally even precise statements are lacking. For details see e.g. McCleary’s “User’s guide to spectral sequences”, Hatcher’s chapter on spectral sequences (available online), or for a purely algebraic viewpoint, Weibel’s text on homological algebra. There is an elegant abstract construction of spectral sequences based on the notion of an “exact couple”. I won’t discuss exact couples here—in fact, I won’t even give the abstract definition of a spectral sequence—because the abstract view is easy (I’ll leave it to you to read about it), whereas really learning spectral sequences requires working with the details in specific cases. The essay is not supposed to be read in linear order. For instance, when reading the section on the spectral sequence of a filtered space, you will want to simultaneously read about the Serre spectral sequence to see an example. It’s essential to draw diagrams of the E2-page of your spectral sequence, for example dots corresponding to integer lattice points in the first quadrant, with each dot representing a module over some base ring R (for us, R is a principal ideal domain) and with arrows between dots representing the differentials of the spectral sequence. I haven’t had time to include such diagrams here, but will provide plenty of them in class.