Zeeman's comparison theorem

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 $$\mathbb{Q}$$ as coefficient ring, we have the Serre spectral sequence $$E_2^{p,q}$$ for the fibration $$G \to EG \to BG$$. We have: $$E_{\infty} \simeq \mathbb{Q}$$ since EG is contractible. We also have a theorem of Hopf stating that $$H^*(G; \mathbb{Q}) \simeq \Lambda(u_1, \dots, u_n)$$, an exterior algebra generated by finitely many homogeneous elements.

Next, we let $$E(i)$$ be the spectral sequence whose second page is $$E(i)_2 = \Lambda(x_i) \otimes \mathbb{Q}[y_i]$$ and whose nontrivial differentials on the r-th page are given by $$d(x_i) = y_i$$ and the graded Leibniz rule. Let $${}^{\prime} E_{r} = \otimes_i E_{r}(i)$$. Since the cohomology commutes with tensor products as we are working over a field, $${}^{\prime} E_{r}$$ is again a spectral sequence such that $${}^{\prime} E_{\infty} \simeq \mathbb{Q} \otimes \dots \otimes \mathbb{Q} \simeq \mathbb{Q}$$. Then we let
 * $$f: {}^{\prime} E_r \to E_r, \, x_i \mapsto u_i.$$

Note, by definition, f gives the isomorphism $${}^{\prime} E_r^{0, q} \simeq E_r^{0, q} = H^q(G; \mathbb{Q}).$$ 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: $$E_2^{p, 0} \simeq {}^{\prime} E_2^{p, 0}$$ as ring by the comparison theorem; that is, $$E_2^{p, 0} = H^p(BG; \mathbb{Q}) \simeq \mathbb{Q}[y_1, \dots, y_n].$$