Local systems in diffeology
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910482402639872 |
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| author | Kuribayashi, Katsuhiko |
| author_facet | Kuribayashi, Katsuhiko |
| contents | By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau--de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13084 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Local systems in diffeology Kuribayashi, Katsuhiko Algebraic Topology Differential Geometry K-Theory and Homology 18F15, 58A35, 58A40, 55U10, 55P62, 55T99 By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau--de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout. |
| title | Local systems in diffeology |
| topic | Algebraic Topology Differential Geometry K-Theory and Homology 18F15, 58A35, 58A40, 55U10, 55P62, 55T99 |
| url | https://arxiv.org/abs/2108.13084 |