In 1967, F. B. Fuller introduces a remarkable index for counting periodic orbits of smooth flows. It has become apparent in recent work of J. Ize and E. N. Dancer that the natural setting for Fuller's index is $\mathbb {T}$-equivariant homotopy theory, where $\mathbb {T}$ is the circle group. This paper describes their work in the conventionnal framework of equivqriqnt stable homotopy theory over a base and index theory for fixed-points of map and zeroes of vector fields.