On multiplicative spectral sequences for nerves and the free loop spaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910450790170624 |
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| author | Kuribayashi, Katsuhiko |
| author_facet | Kuribayashi, Katsuhiko |
| contents | We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a topological category. By applying the machinery to a Borel construction, we determine explicitly the mod $p$ cohomology algebra of the free loop space of the real projective space for each odd prime $p$. This is highlighted as an important computational example of such a spectral sequence. Moreover, we try to represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold $M$ with differential forms on the universal cover of $M$ via Chen's iterated integral map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09827 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On multiplicative spectral sequences for nerves and the free loop spaces Kuribayashi, Katsuhiko Algebraic Topology K-Theory and Homology 55T05, 55T20, 55N25, 55P35, 58A40, 58A10 We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a topological category. By applying the machinery to a Borel construction, we determine explicitly the mod $p$ cohomology algebra of the free loop space of the real projective space for each odd prime $p$. This is highlighted as an important computational example of such a spectral sequence. Moreover, we try to represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold $M$ with differential forms on the universal cover of $M$ via Chen's iterated integral map. |
| title | On multiplicative spectral sequences for nerves and the free loop spaces |
| topic | Algebraic Topology K-Theory and Homology 55T05, 55T20, 55N25, 55P35, 58A40, 58A10 |
| url | https://arxiv.org/abs/2301.09827 |