Lusternik-Schnirelmann category of 𝐒𝐩𝐢𝐧(9)

Iwase, Norio; Kono, Akira

doi:10.1090/S0002-9947-06-04120-1

Lusternik-Schnirelmann category of $\mathbf {Spin}{(9)}$
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by Norio Iwase and Akira Kono PDF

Trans. Amer. Math. Soc. 359 (2007), 1517-1526 Request permission

Abstract:

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

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