Turning vector bundles

Crowley, Diarmuid; Nagy, Csaba; Sims, Blake; Yang, Huijun

doi:10.2140/agt.2024.24.2807

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Turning vector bundles

Diarmuid Crowley, Csaba Nagy, Blake Sims and Huijun Yang

Algebraic & Geometric Topology 24 (2024) 2807–2849

DOI: 10.2140/agt.2024.24.2807

Abstract

We define a turning of a rank- $2 k$ vector bundle $E \to B$ to be a homotopy of bundle automorphisms $ψ_{t}$ from $1_{E}$ , the identity of $E$ , to $- 1_{E}$ , minus the identity, and call a pair $(E, ψ_{t})$ a turned bundle. We investigate when vector bundles admit turnings and develop the theory of turnings and their obstructions. In particular, we determine which rank- $2 k$ bundles over the $2 k$ –sphere are turnable.

If a bundle is turnable, then it is orientable. In the other direction, complex bundles are turned bundles and for bundles over finite CW–complexes with rank in the stable range, Bott’s proof of his periodicity theorem shows that a turning of $E$ defines a homotopy class of complex structure on $E$ . On the other hand, we give examples of rank- $2 k$ bundles over $2 k$ –dimensional spaces, including the tangent bundles of some $2 k$ –manifolds, which are turnable but do not admit a complex structure. Hence turned bundles can be viewed as generalisations of complex bundles.

We also generalise the definition of turning to other settings, including other paths of automorphisms, and we relate the generalised turnability of vector bundles to the topology of their gauge groups and the computation of certain Samelson products.

Keywords

vector bundle, complex structure, gauge group, Samelson product

Mathematical Subject Classification

Primary: 57R22

Secondary: 55R15, 55R25

References

Publication

Received: 21 November 2022

Accepted: 4 June 2023

Published: 19 August 2024

Authors

Diarmuid Crowley
School of Mathematics & Statistics University of Melbourne Parkville, VIC Australia

Csaba Nagy
School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom

Blake Sims
School of Mathematics and Statistics University of Melbourne Parkville, VIC Australia

Huijun Yang
School of Mathematics and Statistics Henan University Kaifeng, Henan China

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Volume 24, issue 5 (2024)