Duality functors for triple vector bundles

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.

💡 Research Summary

The paper revisits the dualization operations on triple vector bundles (TVBs) and provides a rigorous categorical framework that corrects an error in the classical treatment by Mackenzie. A TVB is a geometric object equipped with three compatible double‑vector‑bundle structures, one along each of three distinguished directions. Each direction admits a dualization operation, traditionally denoted by a “dual” functor, and the collection of all possible compositions of these three basic dualizations generates a group of symmetries of the TVB.

The authors first observe that Mackenzie’s original calculation, which claimed the dualization group has order 72, implicitly identified certain functors that are not naturally isomorphic. To avoid this pitfall, they reinterpret every dualization as a functor on the category 𝔗𝔙𝔅 of triple vector bundles and linear morphisms, and they declare two functors equal precisely when there exists a natural isomorphism between them. This categorical viewpoint makes it possible to keep track of all coherence data and to recognise hidden identifications that were missed in earlier work.

With this definition in place, the authors list three elementary dualization functors, one for each axis, together with the identity functor. They then write down the fundamental relations that hold up to natural isomorphism: (i) the dualizations along distinct axes commute, (ii) each elementary dualization has period four (four successive applications are naturally isomorphic to the identity), and (iii) certain mixed compositions satisfy cubic relations reminiscent of the symmetric group S₄. By taking the free group generated by the three elementary dualizations and quotienting by the normal subgroup generated by these relations, they obtain a concrete presentation of the dualization group G.

A careful analysis of the resulting presentation shows that G contains 96 distinct equivalence classes of functors. Moreover, the authors prove that G fits into a short exact sequence

  1 → V₄ → G → S₄ → 1,

where V₄ is the Klein four‑group and S₄ is the symmetric group on four letters. The extension is non‑split: G is not a direct product V₄ × S₄ but rather a non‑trivial semidirect product in which S₄ acts on V₄ via a non‑trivial 2‑cocycle. This explains why the order is larger than Mackenzie’s 72; the missing 24 elements correspond precisely to the non‑trivial interaction between the Klein subgroup and the S₄ symmetry.

The paper also discusses the implications of this result. First, it settles a long‑standing ambiguity in the literature concerning the algebraic structure governing dualizations of TVBs. Second, the categorical method—defining dualizations as functors and using natural isomorphisms as the notion of equality—provides a clean, conceptual tool that can be applied to higher‑order structures. The authors sketch how the same approach extends to n‑fold vector bundles: the basic dualizations become n functors, the relations generalise in a straightforward way, and the resulting dualization group will be a non‑split extension of Sₙ by an appropriate elementary abelian 2‑group.

In conclusion, the authors demonstrate that the dualization group of triple vector bundles is a 96‑element non‑split extension of S₄ by the Klein four‑group, correcting the earlier 72‑element claim. Their functorial perspective not only clarifies the algebraic picture for TVBs but also sets the stage for systematic investigations of dualities in higher‑dimensional vector bundle theory.