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Symmetry groups of four-manifolds

Symmetry groups of four-manifolds,10.1016/S0040-9383(01)00006-4,Topology,Michael P. McCooey

Symmetry groups of four-manifolds   (Citations: 14)
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If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply connected four-manifold M with b2(M)⩾3, then it must be isomorphic to a subgroup of S1×S1, 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.
Journal: Topology , vol. 41, no. 4, pp. 835-851, 2002
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    • ...35 any element of π1(M) preserves the orientation of ˜ M ∗ . For the homomorphism α ∗ : G → Aut(H2( ˜ M ∗ ; Z)), by [Mc1] the kernel G0 = ker(¯ α) is an abelian subgroup of T 2 with non-empty fixed point set...

    Fuquan Fanget al. Collapsed 5-manifolds with pinched positive sectional curvature

    • ...In this case the possible finite groups which admit an action are very restricted; in particular, no nonabelian simple groups occur. The following is the main result of [Mc1]...
    • ...Theorem 1. ([Mc1]) Let G be a finite group with a homologically trivial action on a closed 4-manifold M with trivial first homology...
    • ...Proof. Let g be a central involution in S. The possible fixed point sets of g are listed in Lemma 1. If the fixed point set Fix(g) consists of a 2-sphere S 2 and two isolated points then the 2-sphere S 2 is invariant under S. By a result of Edmonds [E], see also [Mc1, Theorem 2], the action on S 2 is orientation preserving...
    • ...dihedral groups has been analyzed in [Mc1], in particular it follows from [Mc1, Prop.13] that in the case b2(M) = 2 a dihedral group has to act without fixed points on M. This contradiction excludes q = 8 and finishes the proof of Lemma 3...
    • ...dihedral groups has been analyzed in [Mc1], in particular it follows from [Mc1, Prop.13] that in the case b2(M) = 2 a dihedral group has to act without fixed points on M. This contradiction excludes q = 8 and finishes the proof of Lemma 3...
    • ...Finally, if g fixes pointwise two 2-spheres then each of these 2-sphere is invariant under (Z3) 2 and (Z3) 2 has two global fixed points on it. By [Mc1, Prop.14], the singular set of (Z3) 2 consists...
    • ...acts orientation-preservingly on it (by [Mc1, Theorem 2]), any involution has to act orientation-preservingly on such a 2-sphere since the action is homologically trivial)...
    • ...Now again a subgroup (Z2) 2 has a global fixed point, and a contradiction to [Mc1, Prop.14] is obtained as in the previous case of the subgroup (Z3) 2 of PSL(2,9)...
    • ...By [Mc1, Prop.14] the singular set of (Z2) 2 is a union of four 2-spheres which is a contradiction since each involution in (Z2) 2 fixes...
    • ...If g fixes a 2-sphere and two isolated points then again a subgroup (Z2) 2 has four fixed points which is a contradiction to [Mc1, Prop.14] since...
    • ...Now G = A5 has a subgroup A4 � (Z2×Z2)⋉Z3, and we consider the normal subgroup (Z2) 2 of A4. By [Mc1, Prop.14] either M has the right intersection form or (Z2) 2 has a global fixed point, so we can assume the latter...
    • ...If g fixes pointwise two 2-spheres then each of these 2-spheres is invariant under the action of (Z2) 2 ; moreover by [Mc1, Theorem 2], since (Z2) 2 acts homologically trivial it acts orientation-preservingly on each of these 2-spheres...
    • ...The factor group G0/K acts faithfully on S 2 + . If G0/K is solvable, we get the thesis; otherwise we can suppose that G0/K is isomorphic to A5 because it is the only nonsolvable finite group acting orientation preservingly on the 2-sphere (the action is orientation preserving by a result of Edmonds [E], see also [Mc1, Theorem 2])...
    • ...If Z has rank two, since it has global fixed point set it is described by [Mc1, Prop.14]...
    • ...If an element of Z leaves invariant both components of Fix(h), it acts on both 2-spheres orientation preservingly (by [Mc1, Theorem 2])...
    • ...Any involution of Z0 different from h acts nontrivially and orientation preservingly (again by [Mc1, Theorem 2]) on the 2-spheres, thus it fixes pointwise on each 2-sphere two points, this implies that the subgroup of rank two generated by h and by the other involution has global fixed point set; this 2-rank subgroup is described by [Mc1, Prop.14] and it contains two involutions different from h, one with 0-dimensional fixed point set and ...
    • ...Any involution of Z0 different from h acts nontrivially and orientation preservingly (again by [Mc1, Theorem 2]) on the 2-spheres, thus it fixes pointwise on each 2-sphere two points, this implies that the subgroup of rank two generated by h and by the other involution has global fixed point set; this 2-rank subgroup is described by [Mc1, Prop.14] and it contains two involutions different from h, one with 0-dimensional fixed point set and ...
    • ...In this case Z1 is completely described by [Mc1, Prop...
    • ...Let h be an element with 2-dimensional fixed point set; h is contained in a subgroup A of rank two with global fixed point set. By [Mc1, Prop...
    • ...We recall that by [Mc1, Theorem 2] the action of G on the two 2spheres has to be orientation preserving and A5 is maximal between the finite groups acting orientation preserving on a 2-sphere...
    • ...We recall that by [Mc1, Theorem 2] the action of G0 on the two 2-spheres has to be orientation preserving and A5 is maximal between the finite groups acting orientation preserving...

    Mattia Mecchiaet al. On finite simple and nonsolvable groups acting on closed 4–manifolds

    • ...[13]). So we may assume that N is homeomorphic to S2. From now on, we shall...

    Jin Hong Kimet al. On nonnegatively curved 4-manifolds with discrete symmetry

    • ... Theorem 2.3 ([21]). Let M be a simply connected compact 4-manifold...

    Fuquan Fang. Finite isometry groups of 4-manifolds with positive sectional curvatur...

    • ...This paper can be viewed as a sequel to [11], where, following a conjecture of Edmonds [8], we showed that if M is a simply-connected four-manifold with b2(M) ≥ 3, and G is a finite or compact Lie group which acts effectively, locally linearly, and homologically trivially on M, then G must be isomorphic to a subgroup of S 1 × S 1 ...
    • ...The analysis of [11] was based on a comparison of the Borel equivariant cohomology of M with that of its singular set �, and an important ingredient in understanding � was Edmonds’s observation [7] that in the simply-connected case, the fixed-point set of any cyclic group action consists only of isolated points and spheres, with no surfaces of higher genus...
    • ...Finally, in [11], gaps in odd degrees frequently led the spectral sequences involved in computing H ∗...
    • ...In [11], we analyzed minimal nonabelian groups and saw in particular that if G is a nonabelian rank one finite group such that every proper subgroup of G is abelian, then G is either a metacyclic group of the form Cp ⋊ Cqn, where p and q are prime, and Cqn acts on Cp via an order q group automorphism, or Q8, the order 8 quaternion group...

    Michael McCooey. Symmetry groups of non-simply-connected four-manifolds

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