Vol. 95, No. 2, 1981

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Smooth actions of the circle group on exotic spheres

V. J. Joseph

Vol. 95 (1981), No. 2, 323–336

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.

Mathematical Subject Classification 2000
Primary: 57S25
Received: 10 April 1979
Published: 1 August 1981
V. J. Joseph