Fixed Point Free Involutions and Equivariant Maps

1. Preliminaries. We are concerned with involutions without fixed points, together with equivariant maps connecting such involutions. An involution T is a homeomorphism of period 2 of a Hausdorff space X onto itself; that is, T(x) = x for all x £ X . There is associated with an involution T on X the orbit space X/T, obtained by identifying x with T(x) for all x G Z . Denote by v\ X—+X/T the decomposition map. If T is fixed point free, then v\ X—*X/Tis a local homeomorphism. The map v is always both open and closed. In addition to fixed point free involutions we also study equivariant maps. If spaces X, X carry involutions T, V respectively, then a map m-.X—^X' is equivariant provided that m(T(x)) — T(m(x)) for all xÇzX. An equivariant map m: X-+X' induces a map M: X/T -*X'/T'. The most fundamental involution without fixed points is the antipodal involution A on the ^-sphere 5, given by A(xi> • • • , xn+i) = ( — #i, • • * , — Xn+i). When we speak of S as carrying a fixed point free involution, it is to be understood that we refer to A. For 5 n , we have the classical body of results of Lyusternik-Schnirelmann and Borsuk-Ulam. Some of these well-known results are summarized in the following. (1.1) The following are true for every n: (i) there is no equivariant map of 5 n + 1 into S; (ii) every equivariant map of S into itself is of odd degree; in particular, it is essential] (iii) the Lyusternik-Schnirelmann category of S/A =P W is n; (iv) for every covering of S by n + 1 closed sets Ai, • • • , An+i9 some set Ai contains an antipodal pair; (v) for every map ƒ : S—>R there is a point x £ S n for which f(x) =f{A{x)).