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arXiv · 2306.11685

$W$-triviality of low dimensional manifolds

Aritra C Bhattacharya·Bikramjit Kundu·Aniruddha C Naolekar
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Abstract

A space $X$ is $W$-trivial if for every real vector bundle $\alpha$ over $X$ the total Stiefel-Whitney class $w(\alpha)$ 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.

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Aritra C Bhattacharya, Bikramjit Kundu, Aniruddha C Naolekar. 2023-06-20. $W$-triviality of low dimensional manifolds. https://doi.org/10.1007/s00229-024-01575-x

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