Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913479573635072 |
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| author | Matmat, Chahrazade Blanchet, Christian |
| author_facet | Matmat, Chahrazade Blanchet, Christian |
| contents | We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a compact connected 4-dimensional manifold with free involution. This $\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists an equivariant map with target the $n$-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds Matmat, Chahrazade Blanchet, Christian Geometric Topology Algebraic Topology 57M27, 57M60 We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a compact connected 4-dimensional manifold with free involution. This $\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists an equivariant map with target the $n$-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction. |
| title | Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds |
| topic | Geometric Topology Algebraic Topology 57M27, 57M60 |
| url | https://arxiv.org/abs/2401.15412 |