Groups that act pseudofreely on S<sup>2</sup>×S<sup>2</sup>

McCooey, Michael P.

doi:10.2140/pjm.2007.230.381

Cited by 3 publications

(5 citation statements)

References 14 publications

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“…So each orbit under (Z 2 ) 3 of a fixed point of an involution has exactly four elements, and there are exactly seven such orbits. This is exactly the situation excluded for a group (Z 2 ) 2 in the proof of [Mc2,Lemma 4.5], so PSL(2, 8) does not act.…”

Section: Proof Of Theoremmentioning

confidence: 74%

“…Again by part (i), this is possible only for q = 2, 4 or 8. If q = 8 then a dihedral subgroup ‫ޚ‬ p ‫ޚ‬ 2 of U has a fixed point in M. The situation for actions of dihedral groups has been analyzed in [McCooey 2002]; in particular it follows from [McCooey 2002, Proposition 13] that in the case b 2 (M) = 2 a dihedral group has to act without fixed points on M. This contradiction excludes q = 8.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Theorem 1 [McCooey 2002]. Let G be a finite group with a homologically trivial action on a closed 4-manifold M with trivial first homology.…”

Section: Introductionmentioning

confidence: 99%

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On finite simple and nonsolvable groups acting on closed 4–manifolds

Mecchia¹,

Zimmermann²

2009

Pacific J. Math.

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We show that the only finite nonabelian simple groups to admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups ‫ށ‬ 5 , ‫ށ‬ 6 and the linear fractional group PSL(2, 7). (We note that for homologically nontrivial actions all finite groups occur.) The situation depends strongly on the second Betti number b 2 (M) of M and was known before if b 2 (M) is different from two, so the main new result concerns the case b 2 (M) = 2. We prove that the only simple group that occurs in this case is ‫ށ‬ 5 , and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

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Section: Proof Of Theoremmentioning

confidence: 74%

Section: Proof Of Theoremmentioning

confidence: 99%

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On finite simple and nonsolvable groups acting on closed 4–manifolds

Mecchia¹,

Zimmermann²

2009

Pacific J. Math.

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“…By Lemmas 3.1, 3.2, 3.4, Theorem 2.4 and [20] for M = S 2 × S 2 , it is straightforward to verify that Let G 0 G be as above satisfying that |G| ≥ 3 10 500 . If M is not homeomorphic to S 4 , then…”

Section: Lemmasmentioning

confidence: 95%

Finite isometry groups of 4-manifolds with positive sectional curvature

Fang

2007

Math. Z.

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Let M be an oriented compact Riemannian 4-manifold with positive sectional curvature. Let G be a finite subgroup of the isometry group of M. We prove that, if G is a finite group of order |G| ≥ 3 10 500 , then (i) G is isomorphic to a subgroup of PU (3) if |G| is odd; (ii) G contains an index at most 2 normal subgroup which is isomorphic to a subgroup of SO(5) or PU (3) if |G| is even, and M is not homeomorphic to S 4 . Moreover, M is homeomorphic to CP 2 if G is non-abelian of odd order.

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“…Note that the actions of G = (Z 2 ) 4 on M = S 4 are not pseudofree (see Breton [3]). In addition, M = CP 2 admits a pseudofree action of G = Z 3 × Z 3 (see Example 6.2), and S 2 × S 2 admits pseudofree actions of Z 2 × Z 2 (see [15]).…”

Section: Remark 13mentioning

confidence: 99%