Finite group actions on 4-manifolds with nonzero Euler characteristic

Abstract

We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi (X)\ne 0$, then there exists an integer $C\ge 1$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ satisfying $[G:A]\le C$, $\chi (X^A)=\chi (X)$, and $A$ can be generated by at most $2$ elements. Furthermore, if $\chi (X)<0$ then $A$ is cyclic. This proves, for any such $X$, a conjecture of Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.

18 pages, v2: the main theorem has been strengthened for manifolds with negative Euler characteristic; a gap has been corrected in the proof of Lemma 6.1 of v1, which in v2 has been split in Lemmas 6.1 and 6.2; part of the introduction has been rewritten; some other minor changes; v3: final version, substantial revision of v2, to appear in Mathematische Zeitschrift

References in corpus (7)

• Groups acting on manifolds: around the Zimmer program
• Spin squeezing in optical lattice clocks via lattice-based QND measurements
• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Finite group actions on manifolds without odd cohomology
• Finite subgroups of diffeomorphism groups
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Cited by in corpus (5)

• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Finite group actions on manifolds without odd cohomology
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Finite groups acting symplectically on $T^2\times S^2$
• On Jordan type bounds for finite groups of diffeomorphisms of 3-manifolds and Euclidean spaces