A note on the Samelson products in \(\pi_*(SO(2n))\) and the group \([SO(2n),SO(2n)]\)
Consider the map \(j\, ':BSO(2n)\to BSO(2n+1)\) with fibre \(\alpha ':S^{2n}\to BSO(2n),\) where \(j\, '=Bj,\) \(j:SO(2n)\to SO(2n+1)\) is the inclusion and \(\alpha '\) is the adjoint of an element \(\alpha \in \pi_{2n-1}(SO(2n)).\) For odd \(p\) the authors obtain elements whose Samelson products with \(\alpha \) are non-zero in \(\pi _*(SO(2n))_{(p)}.\) They use this result to show that the group \([SO(2n),SO(2n)]_{(p)}\) is commutative if, and only if, \(p>4n-5\) (for \(n\neq 3,4,\) or \(6\)) or \(p\geq 4n-5\) (for \(n= 3,4,\) or \(6\)).
scientific article; zbMATH DE number 5121449
scientific article