2005/07/23 by Norio Iwase, Iwase, Norio, Akira Kono +1 · 1 citation
Mathematics · #55M30 #55N20 #57T30 #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:55M30 #msc:55N20 #msc:57T30
paper · pdf · doi:10.48550/arxiv.math/0507491
12pages
arxiv created 2005/07/23 · arxiv updated 2009/12/01
Let G be a compact connected Lie group and p : E → Σ2V a principal G-bundle with a characteristic map α: A=ΣV → G. By combining cone decomposition arguments in Iwase-Mimura-Nishimoto [3,5] with computations of higher Hopf invariants introduced in Iwase [8], we generalize the result in Iwase-Mimura [12]: Let Fi|0 ≤ i ≤ m be a cone-decomposition of G with a canonical structure map σi of cat(Fi) ≤ i for i ≤ m. We have cat(E) ≤ \Max(m+n,m+2) for n ≥ 1, if αis compressible into Fn ⊆ Fm ≃ G and Hσnn(α) = 0, under a suitable compatibility condition. On the other hand, calculations of Hamanaka-Kono [3] and Ishitoya-Kono-Toda [5] on spinor groups yields a lower estimate for the L-S category of spinor groups by means of a new computable invariant Mwgt(-;mathbbF2) which is stronger than wgt(-;\mathbbF2) introduced in Rudyak [16] and Strom [18]. As a result, we obtain cat(Spin(9)) = Mwgt(Spin(9);\mathbbF2) = 8 > 6 = wgt(Spin(9);\mathbbF2).