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Turning vector bundles

Diarmuid Crowley, Csaba Nagy, Blake Sims and Huijun Yang

Algebraic & Geometric Topology 24 (2024) 2807–2849
Abstract

We define a turning of a rank-2k vector bundle E B to be a homotopy of bundle automorphisms ψt from 1E, the identity of E, to 1E, minus the identity, and call a pair (E,ψt) 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-2k bundles over the 2k–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-2k bundles over 2k–dimensional spaces, including the tangent bundles of some 2k–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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