|
This article is cited in 1 scientific paper (total in 1 paper)
On the equivalence of some spectral sequences for Serre fibrations
A. Yu. Onishchenko, F. Yu. Popelenskii M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
Several different constructions of a spectral sequence for a Serre fibration $\pi\colon E \to B$ over a compact simply connected manifold $B$ are considered in this paper. Namely, we consider the spectral sequence for the minimal model $(\Lambda V\otimes \Lambda W,d)$ of the fibration, along with the spectral sequences arising from the Čech filtration in the complexes $\check{C}^*(\mathscr{U}, A_{PL}^*(\pi^{-1}(U)))$ and $\check{C}^*(\mathscr{U}, S^*(\pi^{-1}(U)))$, where $\mathscr{U}=\{U\}$ is a covering of the base $B$. It is known that all these spectral sequences have the same terms $E_2^{*,*}=H^*(X)\otimes H^*(F)$ and converge to the cohomology of the total space $E$. A new natural isomorphism of these spectral
sequences is constructed in every term $E_r$ with $r\ge2$. It is also proved that in the case of a smooth locally trivial fibration these spectral sequences are isomorphic to the spectral sequences of the complex of smooth forms $\Omega^*(E)$ and of the Čech-de Rham complex. It is therefore established that all these
constructions give the same spectral sequence, starting from the $E_2$ term.
Bibliography: 9 titles.
Keywords:
spectral sequences, Serre fibration, Čech-de Rham complex, minimal model.
Received: 08.06.2010
Citation:
A. Yu. Onishchenko, F. Yu. Popelenskii, “On the equivalence of some spectral sequences for Serre fibrations”, Mat. Sb., 202:4 (2011), 85–110; Sb. Math., 202:4 (2011), 547–570
Linking options:
https://www.mathnet.ru/eng/sm7755https://doi.org/10.1070/SM2011v202n04ABEH004155 https://www.mathnet.ru/eng/sm/v202/i4/p85
|
Statistics & downloads: |
Abstract page: | 536 | Russian version PDF: | 202 | English version PDF: | 11 | References: | 54 | First page: | 34 |
|