Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866915034181926912 |
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| author | Neumann, Frank Szymik, Markus |
| author_facet | Neumann, Frank Szymik, Markus |
| contents | The Leray-Serre and the Eilenberg-Moore spectral sequences are fundamental tools for computing the cohomology of a group or, more generally, of a space. We describe the relationship between these two spectral sequences when both of them share the same abutment. There exists a joint tri-graded refinement of the Leray--Serre and the Eilenberg--Moore spectral sequence. This refinement involves two more spectral sequences, the preludes from the title, which abut to the initial terms of the Leray--Serre and the Eilenberg--Moore spectral sequence, respectively. We show that one of these always degenerates from its second page on and that the other one satisfies a local-to-global property: It degenerates for all possible base spaces if and only if it does so when the base space is contractible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_10209 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences Neumann, Frank Szymik, Markus Algebraic Topology Differential Geometry Geometric Topology K-Theory and Homology The Leray-Serre and the Eilenberg-Moore spectral sequences are fundamental tools for computing the cohomology of a group or, more generally, of a space. We describe the relationship between these two spectral sequences when both of them share the same abutment. There exists a joint tri-graded refinement of the Leray--Serre and the Eilenberg--Moore spectral sequence. This refinement involves two more spectral sequences, the preludes from the title, which abut to the initial terms of the Leray--Serre and the Eilenberg--Moore spectral sequence, respectively. We show that one of these always degenerates from its second page on and that the other one satisfies a local-to-global property: It degenerates for all possible base spaces if and only if it does so when the base space is contractible. |
| title | Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences |
| topic | Algebraic Topology Differential Geometry Geometric Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2106.10209 |