The group of self-homotopy classes of SO(4)
Let \({\mathcal H}(G)=[G,G]\) denote the group of homotopy classes of maps of a topological group \(G\) into itself, with group structure induced by that of \(G\). The author considers the case where \(G\) is a compact, connected, semi-simple and non-simple Lie group of rank \(2\), that is \(S^3\times S^3\), \(S^3\times SO(3)\), \(SO(4)\) or \(SO(3)\times SO(3)\). He gives presentations for \({\mathcal H}(G)\) in the cases of \(S^3\times SO(3)\) and \(SO(4)\); results for \(S^3\times S^3\) and certain other low rank Lie groups are already known. He determines the nilpotency of \({\mathcal H}(G)\) in each of the four cases, and shows that \({\mathcal H}(S^3\times SO(3))\) and \({\mathcal H}(SO(4))\) are not isomorphic, even though \(S^3\times SO(3)\) and \(SO(4)\) are homeomorphic.
scientific article; zbMATH DE number 2005927
scientific article