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Compact Lie Group
equivariant cohomology
Finite Group
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Homologically Trivial Group Actions on 4Manifolds
On the classification of finite groups acting on homology 3spheres
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Symmetry groups of fourmanifolds
Symmetry groups of fourmanifolds,10.1016/S00409383(01)000064,Topology,Michael P. McCooey
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Symmetry groups of fourmanifolds
(
Citations: 14
)
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Michael P. McCooey
If a (possibly finite)
compact Lie group
acts effectively, locally linearly, and homologically trivially on a closed, simply connected fourmanifold M with b2(M)⩾3, then it must be isomorphic to a subgroup of S1×S1, and the action must have nonempty fixedpoint 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. 835851, 2002
DOI:
10.1016/S00409383(01)000064
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Citation Context
(7)
...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 nonempty fixed point set...
Fuquan Fang
,
et al.
Collapsed 5manifolds 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 4manifold 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 2sphere S 2 and two isolated points then the 2sphere 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 2spheres then each of these 2sphere 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 orientationpreservingly on it (by [
Mc1
, Theorem 2]), any involution has to act orientationpreservingly on such a 2sphere 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 2spheres which is a contradiction since each involution in (Z2) 2 fixes...
...If g fixes a 2sphere 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 2spheres then each of these 2spheres is invariant under the action of (Z2) 2 ; moreover by [
Mc1
, Theorem 2], since (Z2) 2 acts homologically trivial it acts orientationpreservingly on each of these 2spheres...
...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 2sphere (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 2spheres 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 2spheres, thus it fixes pointwise on each 2sphere two points, this implies that the subgroup of rank two generated by h and by the other involution has global fixed point set; this 2rank subgroup is described by [Mc1, Prop.14] and it contains two involutions different from h, one with 0dimensional fixed point set and ...
...Any involution of Z0 different from h acts nontrivially and orientation preservingly (again by [Mc1, Theorem 2]) on the 2spheres, thus it fixes pointwise on each 2sphere two points, this implies that the subgroup of rank two generated by h and by the other involution has global fixed point set; this 2rank subgroup is described by [
Mc1
, Prop.14] and it contains two involutions different from h, one with 0dimensional fixed point set and ...
...In this case Z1 is completely described by [
Mc1
, Prop...
...Let h be an element with 2dimensional 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 2sphere...
...We recall that by [
Mc1
, Theorem 2] the action of G0 on the two 2spheres has to be orientation preserving and A5 is maximal between the finite groups acting orientation preserving...
Mattia Mecchia
,
et 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 Kim
,
et al.
On nonnegatively curved 4manifolds with discrete symmetry
... Theorem 2.3 ([
21
]). Let M be a simply connected compact 4manifold...
Fuquan Fang
.
Finite isometry groups of 4manifolds 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 simplyconnected fourmanifold 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 simplyconnected case, the fixedpoint 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 nonsimplyconnected fourmanifolds
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Citations
(14)
Collapsed 5manifolds with pinched positive sectional curvature
(
Citations: 4
)
Fuquan Fang
,
Xiaochun Rong
Journal:
Advances in Mathematics  ADVAN MATH
, vol. 221, no. 3, pp. 830860, 2009
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(
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