W-triviality of low dimensional manifolds

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...

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Bibliographic Details
Published in:Manuscripta mathematica 2024-09, Vol.175 (1-2), p.499-512
Main Authors: Bhattacharya, Aritra C., Kundu, Bikramjit, Naolekar, Aniruddha C.
Format: Article
Language:English
Subjects:
Algebraic Geometry
Calculus of Variations and Optimal Control
Optimization
Geometry
Lie Groups
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Number Theory
Topological Groups
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Summary: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.
ISSN:0025-2611
1432-1785
DOI:10.1007/s00229-024-01575-x