On Lusternik-Schnirelmann category of SO(10)

Details On Lusternik-Schnirelmann category of SO(10) On Lusternik-Schnirelmann category of SO(10)

2007-12-21

arxiv.org/abs/0712.3637v1

Mathematics

Algebraic Topology

23 pages

Scientific paper

Let $G$ be a compact connected Lie group and $p:E\to \Sigma A (A=\Sigma A_{0})$ a principal G-bundle with a characteristic map $\alpha:A\to G$. We assume that there is a cone-decomposition $\{K_{i}\to F_{i-1}\to F_{i} | 1\le i \le n, F_{0}= \{\ast\} \text{and}F_{n}\simeq X \}$ of $G$ of length $m$. Our main theorem is as follows: we have $\cat{X} \le m+1$, if the characteristic map $\alpha$ is compressible into $F_{1}$ and the Berstein-Hilton Hopf invariant $H_{1}(\alpha)=0 \in [A, \Omega F_1{\ast}\Omega F_1]$. We also apply it to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to determine the L-S category of $\mathrm{SO}(10)$.

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