On Lusternik–Schnirelmann category of $\mathbf{SO}(10)$

Norio Iwase ¹   ; Kai Kikuchi   ; Toshiyuki Miyauchi ²

¹ Faculty of Mathematics Kyushu University Motooka 744 Fukuoka 819-0395, Japan
² Department of Applied Mathematics Faculty of Science Fukuoka University Fukuoka 814-0180, Japan

Fundamenta Mathematicae, Tome 234 (2016) no. 3, pp. 201-227

Résumé

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 $\operatorname{cat}(E) \leq m + 1$.

Détail

DOI : 10.4064/fm678-11-2015

Keywords: compact connected lie group varsigma principal g bundle characteristic map alpha where varsigma i hookrightarrow mid ast varsigma simeq cone decomposition length varsigma subset subset which satisfy subset homotopy operatorname cat under suitable conditions which determine operatorname cat similar result obtained kono first author determine operatorname cat spin result could yield operatorname cat leq