If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold M with b 2 (M ) ≥ 3, then it must be isomorphic to a subgroup of S 1 × S 1 , and the action must have nonempty fixed-point set.Our results strengthen and complement recent work by Edmonds, Hambleton and Lee, and Wilczyński, among others. Our tools include representation theory, finite group theory, and Borel equivariant cohomology.
We show that the simply-connected four-manifolds which admit locally linear, homologically trivial Z p  Z p actions are homeomorphic to connected sums of qCP 2 and S 2  S 2 (with one exception: pseudofree Z 3  Z 3 actions on the Chern manifold), and also establish an equivariant decomposition theorem.This generalizes results from a 1970 paper by Orlik and Raymond about torus actions, and complements more recent work of Fintushel, Yoshida, and Huck on S 1 actions. In each case, the simply-connected four-manifolds which support such actions are essentially the same. 1991 Mathematics Subject Classi®cation: 57S17, 57S25; 20J06. condition holds near the ®xed-point set of the action. Huck and Puppe [10] subsequently generalized further by removing the restriction on H 1 M.Stated simply, the approach of Huck and Puppe was to study the equivariant cohomology of the singular set of an S 1 action (using earlier techniques of Puppe [13]), and thereby derive a characterization of the possible intersection forms. Related methods were used independently by the author [11] to study actions of ®nite nonabelian groups on four-manifolds. Our methods actually simplify somewhat when the groups are abelian, and we apply them here to prove: If M is a closed four-manifold with H 1 M 0 which admits a locally linear, homologically trivial action by Z p  Z p (with p prime), then the intersection form of M splits as a sum of copies of q1 and 0 1 1 0 . Brought to you by | Purdue University Libraries Authenticated Download Date | 5/30/15 1:50 PM 1. H 2 M contains a class u whose square generates H 4 M, and H 3 G has no 3torsion, or 2. b 2 M 3, then the Borel spectral sequence EM collapses with coe½cients in Z or any ®eld. It follows that H à G M is a free H à G module on b 2 M 2 generators corresponding to generators for H à M.Proof. If u e H 2 M has nonzero square, then, since u 3 0, 0 d 3 u 3 3d 3 uu 2 . But E ÃY 4 3 is a free H à G module generated by u 2 . So if H 3 G has no 3-torsion, then d 3 u must be 0. And then, of course, d 3 u 2 0, as well. Thus E 2 M E 3 M. Since d 5 u 2 2ud 5 u 0, the sequence collapses. M. P. McCooey 496 Brought to you by | Purdue University Libraries Authenticated Download Date | 5/30/15 1:50 PM
A pseudofree group action on a space X is one whose set of singular orbits forms a discrete subset of its orbit space. Equivalently -when G is finite and X is compact -the set of singular points in X is finite. In this paper, we classify all of the finite groups which admit pseudofree actions on S 2 × S 2 . The groups are exactly those that admit orthogonal pseudofree actions on S 2 × S 2 ⊂ ޒ 3 × ޒ 3 , and they are explicitly listed. This paper can be viewed as a companion to a preprint of Edmonds, which uniformly treats the case in which the second Betti number of a fourmanifold M is at least three.
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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
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