On Dold-Whitney's parallelizability of 4-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929566900027392 |
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| author | Bais, Valentina |
| author_facet | Bais, Valentina |
| contents | We present a proof of the fact that a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. The contribution of this note is to outline in detail an argument which is essentially due to R. Kirby, using the classification of $SO(4)$-bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Dold-Whitney's parallelizability of 4-manifolds Bais, Valentina Geometric Topology We present a proof of the fact that a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. The contribution of this note is to outline in detail an argument which is essentially due to R. Kirby, using the classification of $SO(4)$-bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool. |
| title | On Dold-Whitney's parallelizability of 4-manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2410.22117 |