Zeeman's comparison theorem explained

In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman, gives conditions for a morphism of spectral sequences to be an isomorphism.

Illustrative example

As an illustration, we sketch the proof of Borel's theorem, which says the cohomology ring of a classifying space is a polynomial ring.

First of all, with G as a Lie group and with

Q as coefficient ring, we have the Serre spectral sequence E ^{p,q} _{2} for the fibration G \to EG \to BG . We have: E _{infty} \simeq Q since EG is contractible. We also have a theorem of Hopf stating that H ^{*(G;} Q ) \simeq Λ (u _{1,} ... , u _{n)} , an exterior algebra generated by finitely many homogeneous elements.

Next, we let

E(i) be the spectral sequence whose second page is E(i) _{2} = Λ (x _{i)} ⊗ Q [y _{i]} and whose nontrivial differentials on the r-th page are given by d(x _{i)} = y _{i} and the graded Leibniz rule. Let {} ^{\prime} E _{r} = ⊗ _{i} E _{r} (i) . Since the cohomology commutes with tensor products as we are working over a field, {} ^{\prime} E _{r} is again a spectral sequence such that {} ^{\prime} E _{infty} \simeq Q ⊗ ... ⊗ Q \simeq Q . Then we letf: {} ^{\prime} E _{r} \to E _{r,} x _{i} \mapsto u _{i.} Note, by definition, f gives the isomorphism {} ^{\prime} E ^{0 , q} _{r} \simeq E ^{0 , q} _{r} = H ^{q(G;} Q ). A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that u _{i} are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: E ^{p, 0} _{2} \simeq {} ^{\prime} E ^{p, 0} _{2} as ring by the comparison theorem; that is, E ^{p, 0} _{2} = H ^{p(BG;} Q ) \simeq Q [y _{1,} ... , y _{n].}