1), so they are of the form These maps are all zero since they are , hence the spectral sequence degenerates: . Say, it converges to with a filtration such that . Then , , , , etc. Thus, there is the exact sequence: . Next, let be a spectral sequence whose second page consists only of two lines q = 0, 1. This need not degenerate at the second page but it still degenerates at the third page as the differentials there have degree (-3, 2). Note , as the denominator is zero. Similarly, . Thus, . Now, say, the spectral sequence converges to H with a filtration F as in the previous example. Since , , etc., we have: . Putting everything together, one gets: Wang sequence The computation in the previous section generalizes in a straightforward way. Consider a fibration over a sphere: with n at least 2. There is the Serre spectral sequence: ; that is to say, with some filtration . Since is nonzero only when p is zero or n and equal to Z in that case, we see consists of only two lines , hence the -page is given by Moreover, since for by the universal coefficient theorem, the page looks like Since the only non-zero differentials are on the -page, given by which is the spectral sequence converges on . By computing we get an exact sequence and written out using the homology groups, this is To establish what the two -terms are, write , and since , etc., we have: and thus, since , This is the exact sequence Putting all calculations together, one gets: (The Gysin sequence is obtained in a similar way.) Low-degree terms With an obvious notational change, the type of the computations in the previous examples can also be carried out for cohomological spectral sequence. Let be a first-quadrant spectral sequence converging to H with the decreasing filtration so that Since is zero if p or q is negative, we have: Since for the same reason and since . Since , . Stacking the sequences together, we get the so-called five-term exact sequence: Edge maps and transgressions Homological spectral sequences Let be a spectral sequence. If for every q < 0, then it must be that: for r ≥ 2, as the denominator is zero. Hence, there is a sequence of monomorphisms: . They are called the