Maps between classifying spaces of Kac--Moody groups
Kac-Moody groups are obtained from Kac--Moody algebras in a way that generalises the usual construction of Lie groups from finite dimensional Lie algebras. In this paper the authors consider the unitary form of a non-affine rank-two Kac--Moody group \(K\). For these groups, the classifying space \(BK\) is a push-out of classifying spaces of compact Lie groups. The authors use various homotopy theoretic methods to study the homotopy classes of self-maps of \(BK\). The integral cohomology of \(BK\) is used to define the degree of a self-map of \(BK\). The authors construct \textit{generic Adams maps} and show that every self-map of \(BK\) is homotopy equivalent to a generic Adams map. They show that self-maps are determined up to homotopy by their restriction to the classifying space of a maximal torus and the authors give a description of the monoid \([BK,BK]\). While some aspects of the homotopy theory of compact Lie groups generalise to the Kac-Moody groups studied here, there are notable exceptions. In Section~6, it is shown that there are non-isomorphic Kac-Moody groups with homotopically equivalent classifying spaces. This is in contrast to the fact that two connected compact Lie groups are isomorphic if and only if their classifying spaces are homotopy equivalent [\textit{D. Notbohm} J. Lond. Math. Soc., II. Ser. 52, 185-198 (1995; Zbl 0836.55008)]. The authors also show that, for a torus \(T\), the natural map from \(Hom(T,K)\) to \([BT,BK]\) is not necessarily surjective (Remark 3.4). For a compact Lie group, the map from \(Hom(T,K)\) to \([BT,BK]\) is surjective (see [\textit{D. Notbohm}, Maps between classifying spaces, Univ. Goettingen, Diss. (1988; Zbl 0675.55009)].
scientific article; zbMATH DE number 1970566
scientific article