Lusternik-Schnirelmann category of 𝐒𝐩𝐢𝐧(9)

Abstract

First we give an upper bound of c a t ( E ) \mathrm {cat}{(E)} , the L-S category of a principal G G -bundle E E for a connected compact group G G with a characteristic map α : Σ V → G \alpha : {\Sigma }V \to G . Assume that there is a cone-decomposition { F i | 0 ≤ i ≤ m } \{F_{i}\,\vert \,0 \leq i\leq m\} of G G in the sense of Ganea that is compatible with multiplication. Then we have c a t ( E ) ≤ M a x ( m + n , m + 2 ) \mathrm {cat}{(E)} \leq \mathrm {Max}(m{+}n,m{+}2) for n ≥ 1 n \geq 1 , if α \alpha is compressible into F n ⊆ F m ≃ G F_{n} \subseteq F_{m}\simeq G with trivial higher Hopf invariant H n ( α ) H_n(\alpha ) . Second, we introduce a new computable lower bound, M w g t ( X ; F 2 ) \mathrm {Mwgt} {(X; {\mathbb {F}_2}}) for c a t ( X ) \mathrm {cat}({X}) . The two new estimates imply c a t ( S p i n ( 9 ) ) = M w g t ( S p i n ( 9 ) ; F 2 ) = 8 > 6 = w g t ( S p i n ( 9 ) ; F 2 ) \mathrm {cat}({\mathbf {Spin}{(9))}}=\mathrm {Mwgt} ({\mathbf {Spin}{(9)};{\mathbb {F}_2}}) = 8 > 6 =\mathrm {wgt}({\mathbf {Spin}{(9)};{\mathbb {F}_2}}) , where ( w g t − ; R ) (\mathrm {wgt}{-;R}) is a category weight due to Rudyak and Strom.