Fixed point free involutions and equivariant maps. II

1. Introduction.We shall continue our discussion of fixed point free involutions which was begun in [2].We denote by S" the antipodal involution on the n-sphere.For any fixed point free involution on a space X the co-index was defined to be the least integer n for which there is an equivariant map X -* S".We abbreviate this invariant to co-ind X.In this terminology the classical Borsuk theorem states that co-ind S" = n.There are also numerous results (for references, see [2]) which among other things relate co-index to the homology of the quotient space X/T.The main purpose of the present note is the computation of the coindex in several examples in which homotopy, rather than homology, considerations are of primary importance.It should be mentioned that A. S. Svarc has also recently studied the application of homotopy theory to equivariant maps [5]; there is a considerable overlap between his work and our previous paper [2].We consider as in our previous paper the space P(S") of paths on S" which join a given point x to its antipode A(x) = -x together with the natural involution of P(S").It is shown that co-ind P(S") = n for n ^ 1, 2, 4 or 8. Next we consider the space V(S") of unit tangent vectors to S", with its involution (the antipodal map on each fibre), and show that co-ind V(S") = n for n # 1, 3, or 7 and co-ind V(S") = n -1 for n = 1, 3 or 7. We also compute the co-index of involutions on low dimensional projective spaces.The arguments rely on suspension and Hopf invariant theorems, using particularly the results of J. F. Adams [1] on maps of Hopf invariant one.2. The space of paths P(S").We choose a base point xeS" and we let P(S") denote the space of all paths in S" which join x to its antipode -x.A fixed point free involution on P(S") is given by T(p)(t) = -p(l -t), where p(t) is a point in P(S").In this section we show (2.1) Theorem.For n # 1, 2, 4 or 8, co-ind P(S") = n.We showed this for n > 1 and odd in [2, p. 425] and we conjectured this result as the general case.We see first that co-ind P(S") = n by defining an equivariant map m : P(S") -► S" as m(p(t)) = p(l/2) e S".Now we suppose there is an equivariant map mx : P(S") -> Sn_1.We define an