On a conjecture of Ōshima
It is a classical result of G.W. Whitehead that the self homotopy group \([G,G]\) for a compact connected Lie group \(G\) has the nilpotency class bounded from above by \(\roman{cat} G\), the reduced Lusternik-Schnirelmann category of \(G\). \textit{H. Ōshima} [J. Math. Kyoto Univ. 40, No.~1, 177--184 (2000; Zbl 1009.55007)] stated a conjecture about a lower bound for the nilpotency class of \([G,G]\): if \(G\) is simple, then \([G,G]\) is nilpotent of class \(\geq \roman{rank}\,G\). The authors show that the conjecture is true for \(\text{PU}(p)\), the \(p\)-dimensional projective unitary group, for any odd prime \(p\). This is the first example satisfying the conjecture for a group of rank greater than 3.
scientific article; zbMATH DE number 5597559
scientific article