On Lusternik-Schnirelmann category of SO(10)

Abstract

Let $G$ be a compact connected Lie group and $p:E\to \Sigma A$ be a principal G-bundle with a characteristic map $\alpha :A\to G$, where $A=\Sigma A_0$ for some $A_0$. Let $\{K_i\to F_{i-1}\hookrightarrow F_i|1\le i\le n,F_0=\{\ast \}{F}_1=\Sigma K_1\text{and}{F}_n\simeq G\}$ be a cone-decomposition of $G$ of length $m$ and $F_1^{\prime }=\Sigma K_1^{\prime }\subset F_1$ with $K_1^{\prime }\subset K_1$ which satisfy $F_i{F}_1^{\prime }\subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\mathrm{cat}(X)\le m+1$, if firstly the characteristic map $\alpha$ is compressible into $F_1^{\prime }$, secondly the Berstein-Hilton Hopf invariant $H_1(\alpha )$ vanishes in $[A,\Omega F_1^{\prime }\ast \Omega F_1^{\prime }]$ and thirdly $K_m$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow \mathrm{SO}(10)\to S^9$ to determine L-S category of $\mathrm{SO}(10)$.

arXiv comment28 pages, 4 figures

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