Zeeman's comparison theorem

Template:Short description In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman,Template:Sfnp 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.Template:Cn

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: $E_{\mathrm{\infty }}\simeq \mathbb{Q}$ since EG is contractible. We also have a theorem of Hopf stating that $H^{*}(G;\mathbb{Q})\simeq \mathrm{\Lambda }(u_1,\dots ,u_n)$, an exterior algebra generated by finitely many homogeneous elements.

Next, we let $E(i)$ be the spectral sequence whose second page is $E(i{)}_2=\mathrm{\Lambda }(x_i)\otimes \mathbb{Q}[y_i]$ 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 ${}^{\prime }{E}_{\mathrm{\infty }}\simeq \mathbb{Q}\otimes \dots \otimes \mathbb{Q}\simeq \mathbb{Q}$. 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, $E_2^{p,0}=H^p(BG;\mathbb{Q})\simeq \mathbb{Q}[y_1,\dots ,y_n].$

Bibliography

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