Voir la notice de l'article provenant de la source Mathematical Sciences Publishers
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.
Crowley, Diarmuid 1 ; Nagy, Csaba 2 ; Sims, Blake 3 ; Yang, Huijun 4
@article{10_2140_agt_2024_24_2807,
author = {Crowley, Diarmuid and Nagy, Csaba and Sims, Blake and Yang, Huijun},
title = {Turning vector bundles},
journal = {Algebraic and Geometric Topology},
pages = {2807--2849},
publisher = {mathdoc},
volume = {24},
number = {5},
year = {2024},
doi = {10.2140/agt.2024.24.2807},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2807/}
}
TY - JOUR AU - Crowley, Diarmuid AU - Nagy, Csaba AU - Sims, Blake AU - Yang, Huijun TI - Turning vector bundles JO - Algebraic and Geometric Topology PY - 2024 SP - 2807 EP - 2849 VL - 24 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2807/ DO - 10.2140/agt.2024.24.2807 ID - 10_2140_agt_2024_24_2807 ER -
%0 Journal Article %A Crowley, Diarmuid %A Nagy, Csaba %A Sims, Blake %A Yang, Huijun %T Turning vector bundles %J Algebraic and Geometric Topology %D 2024 %P 2807-2849 %V 24 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2807/ %R 10.2140/agt.2024.24.2807 %F 10_2140_agt_2024_24_2807
Crowley, Diarmuid; Nagy, Csaba; Sims, Blake; Yang, Huijun. Turning vector bundles. Algebraic and Geometric Topology, Tome 24 (2024) no. 5, pp. 2807-2849. doi: 10.2140/agt.2024.24.2807
Cité par Sources :