Duality functors for $n$-fold vector bundles

Abstract

Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted $\mathcal{D}{\mathcal{F}}_2$, which is the symmetric group $S_3$ on three symbols. In the case of triple vector bundles the authors proved in a previous paper that the corresponding group $\mathcal{D}{\mathcal{F}}_3$ is an extension of $S_4$ by the Klein four-group. In this paper we show that the group $\mathcal{D}{\mathcal{F}}_n$, for $n$-fold vector bundles, $n\ge 3$, is an extension of $S_{n+1}$ by a certain product of groups of order 2, and show that the centre is nontrivial if and only if $n$ is a multiple of 4. The methods employ an interpretation of duality operations in terms of certain graphs on $(n+1)$ vertices.

30 pages, 11 figures, two tables

Cited by in corpus (1)

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