Four-manifolds which admit {Z_p x Z_p} actions

Abstract

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

Finite isometry groups of 4-manifolds with positive sectional curvature

“…If ρ(G) ∩ ker(h) contains Z 2 ⊕ Z 2 , let F denote the fixed point set of Z 2 ⊕ Z 2 . By the same spectral sequence calculation in [19] with Z 2 -coefficients the homology H * (F; Z 2 ) has rank b 2 (M) + 2, where F has dimension at most 1.…”

Concordance of ℤp× ℤpactions onS4

“…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 .…”

“…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.…”

“…The main result of [7] was that if M is a simply-connected four-manifold admitting an effective, homologically trivial, locally linear action by G D ‫ޚ‬ p ‫ޚ‬ p , where p is prime, then 1 M is equivariantly homeomorphic to a connected sum of standard actions on copies of˙‫ރ‬P 2 and S 2 S 2 with a possibly non-standard action on S 4 . In this note we further examine these non-standard actions on the sphere.…”

Symmetry groups of four-manifolds

“…In [13], we treated the case of Z p × Z p actions, and here we apply one result of that paper to study actions of non-abelian groups which contain a rank two abelian subgroup.…”