Symmetry groups of four-manifolds

McCooey, Michael P.

doi:10.1016/s0040-9383(01)00006-4

Cited by 20 publications

(37 citation statements)

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“…Consequently, the number of isolated fixed points of g must be divisible by 2. To eliminate the case where (t, u, v) = (14, 2, 4), note that in this case ∪ i C i has 4 type (B) components each of which contains a fixed point of g. [19] because H is simple and nonabelian. Hence g has 4 isolated fixed points when |g| = 5.…”

Section: End Of Case (1)mentioning

confidence: 99%

Symmetric symplectic homotopy K 3 surfaces

Chen

Kwasik

2011

Journal of Topology

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Abstract. A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

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Section: End Of Case (1)mentioning

confidence: 99%

Symmetric symplectic homotopy K 3 surfaces

Chen

Kwasik

2011

Journal of Topology

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“…Now suppose that G acts on a (open or closed) n-manifold M ; we denote by Σ the singular set of the G-action (all points in M with nontrivial stabilizer). Crucial for the proofs of Theorems 1 and 2 is the following Proposition 1, see [13], [10,11] for a proof (see also [4, Proposition VII.10.1] for a Tate cohomology version ).…”

Section: The Borel Spectral Sequence Associated To a Group Actionmentioning

confidence: 99%

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

Zimmermann

2010

Rend. Circ. Mat. Palermo

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We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2, p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.

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“…The next theorem, due to McCooey [39], is concerned with locally linear, homologically trivial topological actions by a compact Lie group (for example, a finite group) on a closed 4-manifold. (1) If b 2 (M ) = 2 and F = ∅, then G is isomorphic to a subgroup of S 1 × S 1 .…”

Section: Borel Spectral Sequencementioning

confidence: 99%

“…The above-mentioned property of X α can be further exploited to prove the nonexistence of effective smooth G-actions on X α for a certain kind of finite group G. For instance, suppose that G is of odd order and there are no nontrivial G-actions on the E 8 lattice (for example, G is a p-group with p > 7); then any smooth G-actions on X α must be homologically trivial, and therefore, by a theorem of McCooey [39], G must be abelian of rank at most 2 (cf. Corollary 4.4).…”

Section: Introductionmentioning

confidence: 99%

Symmetries and exotic smooth structures on a K3 surface

Chen

Kwasik

2008

Journal of Topology

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Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and a comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and, as a result, the nonsmoothability of actions induced by a holomorphic automorphism of prime order at least 7 is proved, and the nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 that has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of prime order at least 5 is explicitly determined, provided that the action is homologically nontrivial.Corollary 1.3. Any locally linear topological action induced by an automorphism of a K3 surface of prime order at least 7 is nonsmoothable on X α .Proof. Let g be an automorphism of a K3 surface of prime order p 7. If g is nonsymplectic, then g is not smoothable on X α by Theorem 1.1(1). Suppose that g is a symplectic automorphism. Then, by Nikulin [44], we have p = 7 and, moreover, the action of g is pseudofree with three isolated fixed points. Suppose that g is smoothable on X α . Then, by Theorem 1.1(2) and Lemma 4.5, the trace of the action of g on H 2 (X α ; Z) is at least 8, so that, by the Lefschetz fixed-point theorem (cf. Theorem 3.4), the Euler number of the fixed-point set of g is at least 10. Thus we have a contradiction.Next we turn our attention to smooth involutions, that is, smooth Z 2 -actions on X α . Let g : X α → X α be any smooth involution. Since X α is simply connected, g can be lifted to the spin bundle over X α , where there are two cases:(1) g is of even type, meaning that the order of lifting to the spin bundle is 2; or (2) g is of odd type, meaning that the order of lifting to the spin bundle is 4.

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Symmetry groups of four-manifolds

Cited by 20 publications

References 19 publications

Symmetric symplectic homotopy K 3 surfaces

Symmetric symplectic homotopy K 3 surfaces

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

Symmetries and exotic smooth structures on a K3 surface

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