On Lusternik–Schnirelmann category of $\mathbf{SO}(10)$
Volume 234 / 2016
Abstract
Let $G$ be a compact connected Lie group and $p : E\to \varSigma A$ be a principal $G$-bundle with a characteristic map $\alpha : A\to G$, where $A=\varSigma A_{0}$ for some $A_{0}$. Let $\{K_{i} \to F_{i-1} \hookrightarrow F_{i} \mid 1 \le i \le m\}$ with $F_{0}= \{\ast\}$, $F_{1} = \varSigma{K_{1}}$ and $F_{m}\simeq G $ be a cone-decomposition of $G$ of length $m$ and $F’_{1}=\varSigma{K’_{1}} \subset F_{1}$ with $K’_{1} \subset K_{1}$ which satisfy $F_{i}F’_{1} \subset F_{i+1}$ up to homotopy for all $i$. Then $\operatorname{cat}(E) \le m + 1$, under suitable conditions, which is used to determine $\operatorname{cat}({\bf SO}(10))$. A similar result was obtained by Kono and the first author (2007) to determine $\operatorname{cat}({\bf Spin}(9))$, but that result could not yield $\operatorname{cat}(E) \leq m + 1$.
