Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is proved by Crabb--Sutherland and the second author that the number of $A_n$-types of the gauge groups of $P$ is finite if $n<infty$ and $K$ is a finite complex. We show that the number of $A_infty$-types of the gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely many $P$.
We show a homotopy decomposition of $p$-localized suspension $Sigma M_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $pi$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>dim M/2$. We also discuss non-triviality of $pi_{(p)}$ and $Sigma^inftypi$.
A Lie group is called $p$-regular if it has the $p$-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into $p$-regular $mathrm{SO}(2n)_{(p)}$ is determined, which completes the list of (non)triviality of such Samelson products in $p$-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion $mathrm{SO}(2n-1)tomathrm{SO}(2n)$ in the sense of James at any prime $p$.
The (non)triviality of Samelson products of the inclusions of the spheres into p-regular exceptional Lie groups is completely determined, where a connected Lie group is called p-regular if it has the p-local homotopy type of a product of spheres.
A p-compact group is a mod p homotopy theoretical analogue of a compact Lie group. It is determined the homotopy nilpotency class of a p-compact group having the homotopy type of the $p$-completion of the direct product of spheres.
