# Spectral Sequences II

## Two Stripes

The next thing to address in McCleary is an apparent mistake on p. 9 of section 1.2. Here we again assume a first quadrant spectral sequence converging to a graded vector space . This is mentioned at the beginning of the section, but it’s easy to forget that when a bold-faced and titled example (1.D) seems to be presenting a reset of assumptions, rather than building upon prior discussion. Furthermore, in this example, McCleary seems to be working again with the assumption from example 1.A that for . On the other hand, this can be seen as a consequence of the fact that our spectral sequence is limited to the first quadrant, provided the filtration is finite in the sense of Weibel’s Homological Algebra, p. 123 ( for some ). But then it would be unclear why McCleary took this as an additional assumption rather than as a consequence of prior assumptions in the first case. : /

The new part of this example is the assumption that unless or , so all terms of the spectral sequence are to be found just in two horizontal stripes. In particular is only possibly non-zero in these stripes, and since these correspond to filtration quotients, the filtration takes a special form.

First, we might look at the filtration on where .  Note that the spectral sequence terms that give information about are those along the diagonal line where .  Since , the only place where anything interesting might happen is when this line crosses the -axis, i. e. when . This forces , so the only possible nonzero filtration quotient is working with the assumption that . So on the one hand, we get no interesting filtration of for , but on the other hand we can see exactly what it is from the spectral sequence limit.

Now we treat the case of , where . I find this awkward notation again, preferring to reserve for a pure arbitrary spectral sequence index, but since we are trying to address the mistake in this notation, we should keep it for now. The filtration of this vector space/cohomology is interesting when and , where the quotients are given by Every where else, successive quotients are 0, meaning the filtration looks like… In the filtration on page 9, McCleary puts one of the (possibly) non-trivial quotients at instead of at where it should be.  That’s all I’m saying.

This situation is modeled on a spectral sequence for sphere bundles i.e. bundles where the fibers are spheres of a given dimension. The stripes coincide with the fact that a sphere has nontrivial cohomology only at and . This sort of computation is famous enough that it has a name: the Thom-Gysin sequence (or just Gysin sequence).

As a final remark on section 1.2, McCleary says that the sequence in example 1.C is the Gysin sequence. Example 1.C doesn’t exist, we mean example 1.D : )