Kirill C. H. Mackenzie (University of Sheffield)
k.mackenzie@sheffield.ac.uk
The duality of finite-dimensional vector spaces and vector bundles is involutive: $(E^*)^*\cong E$ canonically. For double vector bundles such as the tangent $TE$ or cotangent $T^*E$ of a vector bundle, the symmetric group $S_3$ acts simply transitively on the various duals which can be formed by combining dualizations in the two directions.
For triple vector bundles, Gracia-Saz and the speaker showed (2009) that the corresponding group of duality functors, $DF_3$, has order 96 and is a nonsplit extension of $S_4$ by the Klein four-group. For $n\geq 4$, they showed (2013) that $DF_n$ is an extension of $S_{n+1}$ by a direct product of groups of order 2. Elements of $DF_n$ may also be represented as certain graphs on $n+1$ vertices and the elements of the kernel are then the Euler graphs; that is, the graphs for which each vertex has even valency, and which therefore admit a path of the kind which Euler showed was not available for the bridges of Königsberg.