A note on fixed point free involutions and equivariant maps

Abstract:

The space $P({S^n})$ of all paths $\omega$ in ${S^n}$ with given initial point $x$ and endpoint $- x$ admits an involution $(T\omega )(t) = - \omega (1 - t)$. With the standard antipodal involution on ${S^{n - 1}}$ an equivariant map $P({S^n}) \to {S^{n - 1}}$ is constructed for $n = 2,4,$, or $8$.