Recent work of Schultz
translates the question of which exotic spheres Sn admit semifree circle actions with
k-dimensional fixed point set entirely to problems in homotopy theory provided
the spheres bound spin manifolds. In this article we study circle actions on
homotopy spheres not bounding spin manifolds and prove, in particular, that
the spin boundary hypothesis can be dropped if (n − k) is not divisible by
128. It is also proved that any ordinary sphere can be realized as the fixed
point set of such a circle action on a homotopy sphere which is not a spin
boundary; some of these actions are not necessarily semi-free. This extends
earlier results obtained by Bredon and Schultz. The Adams conjecture, its
consequences regarding splittings of certain classifying spaces and standard
results of simply-connected surgery are used to construct the actions. The
computations involved relate to showing that certain surgery obstructions
vanish.
