Concordance of ℤ_p× ℤ_pactions onS⁴

Abstract

Smith Theory (see Smith [15] and Bredon [1]) shows that when a cyclic group of prime order p acts on a sphere, the fixed-point set is again a ‫ޚ‬ p -homology sphere.The classical Smith Conjecture (whose solution, involving the work of many mathematicians, was published in 1984, see Bass and Morgan [10]) goes further, stating the fixed-point set of a tame cyclic group action on S 3 is either empty, or an unknotted S 1 . But the natural generalization of the Smith conjecture to higher dimensions is false: Giffen [4], Gordon [5], and Sumners [16] all constructed counterexamples in spheres of dimension 4 and higher. Along with [7], these counterexamples motivate our study.Suppose hgi Š ‫ޚ‬ p acts on X D S 4 , locally linearly and preserving orientation. Fix.g/ is then a ‫ޚ‬ p -homology sphere of dimension 0 or 2; since Fix.g/ is also a submanifold, it must be either S 0 or S 2 . Both are possible, but let us assume Fix.g/ Š S 2 . By local linearity, the quotient map X ! X D X= hgi is a branched covering, and the quotient space is a manifold with the image Fix.g/ of the fixed set embedded as a locally flat submanifold. According to Freedman and Quinn [3, 9.3A] (see also Quinn [12]), Fix.g/ has a normal bundle N . The lift of N back to X is an equivariant tubular neighborhood N .Fix.g// of Fix.g/. Then X nN .Fix.g// has the same boundary, and by Alexander duality, the same integral homology, as S 1 B 3 .Lemma 2

A survey of group actions on 4-manifolds

“…It is a theorem of Kervaire, valid in the smooth, PL, or topological locally flat categories, that all 2-knots in S 4 are slice. McCooey [158] has shown that a 2sphere fixed by a C p action is equivariantly slice, in the topological locally linear category 3.3. Free actions.…”