$W$-triviality of low dimensional manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916397408321536 |
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| author | Bhattacharya, Aritra C Kundu, Bikramjit Naolekar, Aniruddha C |
| author_facet | Bhattacharya, Aritra C Kundu, Bikramjit Naolekar, Aniruddha C |
| contents | A space $X$ is $W$-trivial if for every real vector bundle $α$ over $X$ the total Stiefel-Whitney class $w(α)$ is 1. It follows from a result of Milnor that if $X$ is an orientable closed smooth manifold of dimension $1,2,4$ or $8$, then $X$ is not $W$-trivial. In this note we completely characterize $W$-trivial orientable connected closed smooth manifolds in dimensions $3,5$ and $6$. In dimension $7$, we describe necessary conditions for an orientable connected closed smooth $7$-manifold to be $W$-trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $W$-triviality of low dimensional manifolds Bhattacharya, Aritra C Kundu, Bikramjit Naolekar, Aniruddha C Algebraic Topology 57R20 A space $X$ is $W$-trivial if for every real vector bundle $α$ over $X$ the total Stiefel-Whitney class $w(α)$ is 1. It follows from a result of Milnor that if $X$ is an orientable closed smooth manifold of dimension $1,2,4$ or $8$, then $X$ is not $W$-trivial. In this note we completely characterize $W$-trivial orientable connected closed smooth manifolds in dimensions $3,5$ and $6$. In dimension $7$, we describe necessary conditions for an orientable connected closed smooth $7$-manifold to be $W$-trivial. |
| title | $W$-triviality of low dimensional manifolds |
| topic | Algebraic Topology 57R20 |
| url | https://arxiv.org/abs/2306.11685 |