A Borsuk-Ulam theorem for orthogonal \(T^ k\) and \(Z^ r_ p\) actions and applications
The author first gives, for a compact Lie group G, the notion of Borsuk- Ulam property in the weak (resp. strong) sense A (in short, property IA (resp. IIA)) and then proves that the k-dimensional torus group \(T^ k\) and the p-torus \(Z^ r_ p\) have property IIA, which naturally implies IA. The classical Borsuk-Ulam theorem asserts that \(G=Z_ 2\) has property IA. The author introduces the notion of geometric G-index \(\gamma_ G(X)\) for G-spaces X. This index is used to estimate the number of multiple critical points for functions invariant under a \(T^ k\) or \(Z^ r_ p\) action on a compact smooth manifold without boundary.
actions of products of \({\mathbb{Z}}/p\)
scientific article; zbMATH DE number 4159619
scientific article