[Paper Review] Duality and triple structures

Kirill C. H. Mackenzie|ArXiv.org|Jun 14, 2004

Homotopy and Cohomology in Algebraic Topology6 citations

TL;DR

This paper establishes that the duality operations on triple vector bundles generate a group of order 72, distinct from the dihedral or symmetric groups, by embedding the triple structure into a cotangent triple vector bundle. It generalizes the dihedral symmetry of double vector bundle dualities to higher dimensions, revealing a non-trivial group structure that does not extend to hypercube symmetries, and proposes a framework for duality in n-fold vector bundles with a conjectured relation involving (k+1)-fold products of dualization operations.

ABSTRACT

We first recall the basic theory of double vector bundles and the canonical pairing of their duals introduced by the author and by Konieczna and Urbanski. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle. In particular we show that the dihedral group of the triangle acts on this triple via forms of the isomorphisms R introduced by the author and Ping Xu. We then consider the three duals of a general triple vector bundle and show that the corresponding group is neither the dihedral group of the square nor the symmetry group on four symbols.

Motivation & Objective

• To understand the group structure generated by dualization operations on triple vector bundles.
• To extend the duality theory of double vector bundles—previously governed by the symmetric group S3—to the triple case.
• To show that the three duals of a triple vector bundle and the original bundle form the lower faces of a cotangent (n+1)-fold vector bundle structure.
• To identify the group of symmetries arising from dualization in triple vector bundles, contrasting it with expectations from hypercube or symmetric group symmetries.

Proposed method

• The paper constructs a cotangent triple vector bundle from a given double vector bundle, using the canonical pairing of duals introduced by Mackenzie and by Konieczna and Urbański.
• It identifies the dihedral group of order 6 as the symmetry group acting on the three duals of a double vector bundle via isomorphisms R, generalizing earlier results by Mackenzie and Xu.
• For triple vector bundles, the authors define three dualization operations X, Y, Z along the three axes and analyze their compositions.
• Using computational algebra via GAP, the authors verify that the group generated by X, Y, Z satisfies relations including (XYZ)^4 = I, (YZX)^4 = I, (ZXY)^4 = I, and X^2 = Y^2 = Z^2 = I, yielding a group of order 72.
• The paper embeds the triple vector bundle into a cotangent (n+1)-fold vector bundle, where the original and its three duals form the lower n-faces.
• It conjectures a general relation (X_{i1}...X_{ik})^{k+1} = 1 for n-fold vector bundles, extending the pattern observed in the triple case.

Experimental results

Research questions

• RQ1 What is the group structure generated by dualization operations on a triple vector bundle?
• RQ2 How does the duality of triple vector bundles differ from that of double vector bundles, particularly in terms of symmetry?
• RQ3 Can the duality of n-fold vector bundles be systematically generalized beyond the double case?
• RQ4 Why does the group of dualization operations in the triple case fail to be isomorphic to the symmetric group S4 or the dihedral group of the square?
• RQ5 What is the role of the cotangent (n+1)-fold vector bundle in unifying the duality structure of n-fold vector bundles?

Key findings

• The group of dualization operations on a triple vector bundle has order 72, as verified using GAP computational algebra software.
• This group is not isomorphic to the dihedral group of the square or the symmetric group on four symbols, indicating a more complex symmetry than expected.
• The relations X^2 = Y^2 = Z^2 = I, (XYZ)^4 = I, (YZX)^4 = I, and (ZXY)^4 = I fully characterize the group structure.
• The three duals of a triple vector bundle and the original bundle form the three lower n-faces of a cotangent (n+1)-fold vector bundle.
• The subgroup generated by XYXZ, YZXY, and ZXZY has order 12 and is normal, with quotient isomorphic to S3.
• The paper conjectures that for an n-fold vector bundle, the product of k distinct dualization operations raised to the (k+1)th power yields the identity, generalizing the triple case.