Noncommutativity of the group of self homotopy classes of Lie groups (Q1862075)

Let \(G\) be a topological group and let \([G,G]\) be the group of homotopy classes of self maps of \(G\). The main result of the paper is that if \(G\) is \(\text{SU}(n)\) for \(n>2\), or \(\text{Sp}(n)\) for \(n>1\), or \(\text{Spin}(4n)\) for \(n>1\), and if \(H\) is a central subgroup of \(G\), then \([G/H,G/H]\) is not commutative. This extends some earlier results of two of the authors. Furthermore, for certain Lie groups \(G\), they show that nil \([G,G] >2\).