A G-Lusternik-Schnirelman category of space with an action of a compact lie group

AbstractThis work studies the notion of “minimax invariant” of a G-space. Instead of earlier ideas of the G-genus, G-geometrical index, or G-cohomological indexes [1,2,8,15] we develop a purely topological construction. The main idea follows the original approach of Lusternik and Schnirelman. It is very simple, and has all the properties necessary for the minimax procedure. By using the G-category we can extend the classical Krasnosielski theorem. Namely, we show that the orbit space X/G has Lusternik-Schnirelman category equal to n + 1 if a finite group G acts freely on a Z-cohomological n-sphere X