Estimates of the isovariant Borsuk-Ulam constants of connected compact Lie groups
A \(G\)-equivariant map \(f : X \to Y\) is called \(G\)-isovariant if \(f\) preserves the isotropy subgroups, that is, \(G_{f(x)} = G_x\) for every \(x \in X\). Isovariant maps play crucial roles in equivariant topology and transformation groups. Determining existence or non-existence of isovariant maps and their classification up to isovariant homotopy are important problems in the subject. In the paper under review, the author investigates non-existence of \(G\)-isovariant maps between \(G\)-representation spheres. The isovariant version of the Borsuk-Ulam theorem was first studied by \textit{A. G. Wasserman} in [Topology Appl. 38, No. 2, 155--161 (1991; Zbl 0736.55001)]. A compact Lie group \(G\) is called a Borsuk-Ulam group if the isovariant Borsuk-Ulam theorem holds for \(G\)-representations; namely, if the inequality \[ \dim V- \dim V^G \leq \dim W - \dim W^G \] holds whenever there exists a \(G\)-isovariant map \(f : S(V ) \to S(W)\). It is known that finite solvable groups and compact tori are Borsuk-Ulam groups. However, not much is known about this property for other compact groups. As an approach to this problem, the author defines the isovariant Borsuk-Ulam constant \(c_G\) for a (nontrivial) compact Lie group \(G\) as the supremum of \(c \in \mathbb{R}\) such that \[ c(\dim V - \dim V^G) \leq \dim W- \dim W^G \] holds whenever there exists a \(G\)-isovariant map \(f : S(V ) \to S(W)\). The author discusses some properties of \(c_G\) and establishes lower bounds in case of connected compact Lie groups. This leads us to some Borsuk-Ulam type results for isovariant maps. One of the main results of the paper is: Theorem. Let \(G\) be a compact connected Lie group and \(\{G_i\}\) all simple factors of \(G\). Then \(c_G = \min_i\{c_{G_i} \}\).
scientific article; zbMATH DE number 6996442