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,{{sfnp|Zeeman|1957}} gives conditions for a morphism of spectral sequences to be an isomorphism.

Statement

{{math_theorem|name=Comparison theorem|Let $E^r_{p, q}, {}^{\prime}E^r_{p, q}$ 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:

$f: E_2^{p, 0} \to {}^{\prime} E_2^{p, 0}$ is an isomorphism for every p.
$f: E_2^{0, q} \to {}^{\prime} E_2^{0, q}$ is an isomorphism for every q.
$f: E_{\infty}^{p, q} \to {}^{\prime} E_{\infty}^{p, q}$ 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.{{cn|date=February 2024}}

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, $E_2^{p, 0} = H^p(BG; \mathbb{Q}) \simeq \mathbb{Q}[y_1, \dots, y_n].$

References

Bibliography

{{Citation | last1=McCleary | first1=John | title=A User's Guide to Spectral Sequences | publisher=Cambridge University Press | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-56759-6 |mr=1793722 | year=2001 | volume=58}}
{{Citation | last1=Roitberg | first1=Joseph | last2=Hilton | first2=Peter | title=On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations | doi=10.1093/qmath/27.4.433 |mr=0431151 | year=1976 | journal=The Quarterly Journal of Mathematics |series=Second Series | issn=0033-5606 | volume=27 | issue=108 | pages=433–444| url=http://doc.rero.ch/record/300344/files/27-4-433.pdf }}
{{Citation | last1=Zeeman | first1=Erik Christopher | author1-link=Christopher Zeeman | title=A proof of the comparison theorem for spectral sequences | doi=10.1017/S0305004100031984 |mr=0084769 | year=1957 | journal=Proc. Cambridge Philos. Soc. | volume=53 | pages=57–62}}

Category:Spectral sequences

Category:Theorems in algebraic topology