Stable diffeomorphism classification of some unorientable 4-manifolds
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913577430941696 |
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| author | Debray, Arun |
| author_facet | Debray, Arun |
| contents | Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of $S^2\times S^2$, to a series of bordism questions. We implement this in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which $π_1(M)$ is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and four stable diffeomorphism classes for which M is neither. We also determine the corresponding stable homeomorphism classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_03965 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stable diffeomorphism classification of some unorientable 4-manifolds Debray, Arun Geometric Topology Algebraic Topology 57K40 (Primary) 57R75, 57R90 (Secondary) Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of $S^2\times S^2$, to a series of bordism questions. We implement this in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which $π_1(M)$ is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and four stable diffeomorphism classes for which M is neither. We also determine the corresponding stable homeomorphism classes. |
| title | Stable diffeomorphism classification of some unorientable 4-manifolds |
| topic | Geometric Topology Algebraic Topology 57K40 (Primary) 57R75, 57R90 (Secondary) |
| url | https://arxiv.org/abs/2102.03965 |