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An Equivariant Sphere Theorem

Let M be a connected 3-manifold acted on by a group G. Suppose M has a triangulation invariant under G. In this paper it is shown that if there exists an embedded 2-sphere S which does not bound a 3-ball, then there exists such an S for which gS = S or gS 0 S = 0 for every g e G. This result was proved by Meeks, Simon and Yau [3] using analytic techniques. The proof given here is self-contained and elementary. The proof involves looking at embedded 2-spheres which are in general position with respect to the given triangulation. Such a sphere is called minimal if it does not bound a 3-ball and the number of intersections with the 1-skeleton of the triangulation is the smallest possible. The key result proved in this paper is that given a finite set of minimal spheres satisfying a general position condition, there is a finite set of 'standard' disjoint minimal spheres whose union has the same intersection with the 1-skeleton as the original spheres. The set of disjoint spheres is unique up to a homeomorphism of M which fixes the 2-skeleton. In §4 it is shown that if G\K is finite then there is a G-equivariant decomposition of M with irreducible factors. We are then able to deduce from the ordinary loop theorem an equivariant version of the projective plane theorem. In §5 the arguments of the previous sections are modified to provide a proof of the equivariant loop theorem [2]. I think that many of the topological results obtained using analytic minimal surface theory can also be derived using the techniques of this paper. I am grateful to Andrew Bartholomew for pointing out an error in an earlier version of this paper. I thank both Peter Scott and the referee for their helpful comments.