The weak isovariant Borsuk-Ulam theorem for compact Lie groups
The author presents some generalizations of the famous Borsuk-Ulam theorem. For a group \(G\), the weak isovariant Borsuk-Ulam property (WIB-property) in the \({\mathcal F}_G\) is defined to be: if there exists a monotonely increasing function \(\phi_G\) from the set of non-negative integers to itself diverging to \(\infty\) such that for any \(M, N \in {\mathcal F}_G\), \(\phi_G(\dim M - \dim M^G)\leq \dim N -\dim N^G\), provided that there is a \(G\)-isovariant map from \(M\) to \(N\). Here, \({\mathcal F}_G\) may be the family of linear \(G\)-spheres, semi-linear \(G\)-spheres or semi-linear \(G\)-homology spheres. If \(\phi_G\) can be taken to be the identity map, \(G\) is said to have the isovariant Borsuk-Ulam property (IB-property). One of the main results in this paper is: any compact Lie group \(G\) has the WIB-property in the family of linear \(G\)-spheres. The other is that in the family of semi-linear \(G\)-spheres or semi-linear \(G\)-homology spheres, \(G\) has the WIB-property if and only if \(G\) has the IB-property if and only if \(G\) is solvable.
scientific article; zbMATH DE number 2051612