# 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)

`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.