arXiv subjects
Publications and source records attributed to Akira Kono.
Let $\mathcal{G}_{k,n}$ be the gauge group of the principal $\mathrm{Sp}(n)$-bundle over $S^4$ corresponding to $k\in\mathbb{Z}\congπ_3(\mathrm{Sp}(n))$. We refine the result of Sutherland on the homotopy types of $\mathcal{G}_{k,n}$ and relate it with the order of a certain Samelson product in $\mathrm{Sp}(n)$. Then we classify the $p$-local homotopy types of $\mathcal{G}_{k,n}$ for $(p-1)^2+1\ge 2n$.
Let G be a compact connected Lie group and p : E \to Σ^2V a principal G-bundle with a characteristic map α: A=ΣV \to 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 {F_{i}|0 \leq i \leq m} be a cone-decomposition of G with a canonical structure map σ_{i} of cat(F_{i}) \leq i for i \leq m. We have cat(E) \leq \Max(m+n,m+2) for n \geq 1, if αis compressible into F_{n} \subseteq F_{m} \simeq G and H^{σ_n}_n(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(-;{mathbb{F}_2}) which is stronger than wgt(-;{\mathbb{F}_2}) introduced in Rudyak [16] and Strom [18]. As a result, we obtain cat(Spin(9)) = Mwgt(Spin(9);\mathbb{F}_2) = 8 > 6 = wgt(Spin(9);\mathbb{F}_2).