Zeeman's comparison theorem

Short description: On when a morphism of spectral sequences in homological algebra is an isomorphism

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

Statement

Comparison theorem — Let $E_{p,q}^r,{}^{\prime }{E}_{p,q}^r$ be first quadrant spectral sequences of flat modules over a commutative ring and $f:E^r\to {}^{\prime }{E}^r$ a morphism between them. Then any two of the following statements implies the third:

1. $f:E_2^{p,0}\to {}^{\prime }{E}_2^{p,0}$ is an isomorphism for every p.
2. $f:E_2^{0,q}\to {}^{\prime }{E}_2^{0,q}$ is an isomorphism for every q.
3. $f:E_{\mathrm{\infty }}^{p,q}\to {}^{\prime }{E}_{\mathrm{\infty }}^{p,q}$ is an isomorphism for every p, q.

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

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

$f:{}^{\prime }{E}_r\to E_r,x_i\mapsto u_i.$

Note, by definition, f gives the isomorphism ${}^{\prime }{E}_r^{0,q}\simeq E_r^{0,q}=H^q(G;\mathbb{\mathbb{Q}}).$ 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, $E_2^{p,0}=H^p(BG;\mathbb{\mathbb{Q}})\simeq \mathbb{\mathbb{Q}}[y_1,\dots ,y_n].$

Bibliography

• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6 
• Roitberg, Joseph; Hilton, Peter (1976), "On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations", The Quarterly Journal of Mathematics, Second Series 27 (108): 433–444, doi:10.1093/qmath/27.4.433, ISSN 0033-5606, http://doc.rero.ch/record/300344/files/27-4-433.pdf 
• Zeeman, Erik Christopher (1957), "A proof of the comparison theorem for spectral sequences", Proc. Cambridge Philos. Soc. 53: 57–62, doi:10.1017/S0305004100031984