Duality functors for triple vector bundles

TL;DR

This paper computes the dualization operation group for triple vector bundles, revealing it has order 96, and interprets these operations as functors, with implications for n-fold vector bundles.

Contribution

It corrects the order of the dualization group for triple vector bundles and provides a functorial interpretation, extending previous work by Mackenzie.

Findings

The dualization group has order 96, not 72.

The group is a nonsplit extension of S4 by the Klein group.

Methodology will be applied to n-fold vector bundles in future work.

Abstract

We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equal if they are naturally isomorphic. The method set out here will be applied in a subsequent paper to the case of n-fold vector bundles.