Theory of connexes. II (Q1068740)

[For part I see ibid. 17, 777-812 (1981; Zbl 0476.90100).] - Considered is the famous game named Hex, where two players occupy the vertices in a rhombus and who obtains a path between his initially posed pieces wins. It is remarkable that this game always gives a single winner. Regarding the board as the upper half of the sphere, we notice the following statement: ''Suppose there is a simplicial decomposition of the sphere invariant by the antipodal mapping. If two players occupy all the dipoles of vertices, then there exists strictly one player who obtains in his territory a connected set of vertices invariant under the antipodal action.'' Our purpose in this paper is to prove the above statement in a more general situation.