2012/08/31 by Alfonso Gracia-Saz, Gracia-Saz, Alfonso, K. C. H. Mackenzie +1
Mathematics · #18D05 #18D35 #20E99 #53D17 (primary) #55R99 (secondary) #Category Theory (math.CT) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #math.CO #math.CT #math.DG #msc:18D05 #msc:18D35 #msc:20E99 #msc:53D17 #msc:55R99
paper · pdf · doi:10.48550/arxiv.1209.0027
30 pages, 11 figures, two tables
arxiv created 2012/08/31 · arxiv updated 2012/09/04
Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted \mathscrD\mathscrF2, which is the symmetric group S3 on three symbols. In the case of triple vector bundles the authors proved in a previous paper that the corresponding group \mathscrD\mathscrF3 is an extension of S4 by the Klein four-group. In this paper we show that the group \mathscrD\mathscrFn, for n-fold vector bundles, n≥ 3, is an extension of Sn+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.