Manchester Geometry Seminar 2012/2013

Thursday 21 March 2013. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

Multiple Duality and the Bridges of Königsberg

Kirill C. H. Mackenzie (University of Sheffield)

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.