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Finite group actions on 4-manifolds with nonzero Euler characteristic

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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 a natural number C such that any finite group G acting smoothly and effectively on X has an abelian subgroup A generated by two elements which satisfies \([G:A]\le C\) and \(\chi (X^A)=\chi (X)\). Furthermore, if \(\chi (X)<0\) then A is cyclic. This answers positively, for any such X, a question of Étienne Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.

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Notes

  1. I thank A. Jaikin and E. Khukhro for explaining this argument to me.

  2. However, for some restricted classes of groups acting on X one can study in detail the topology of the singular set; in the case of minimal non-abelian groups, this is done in [15, 16], and it is the crucial ingredient of the proofs.

  3. See [10, §1.4] for the definition of neat submanifold.

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  1. Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Ignasi Mundet i Riera

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  1. Ignasi Mundet i Riera
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Correspondence to Ignasi Mundet i Riera.

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This work has been partially supported by the (Spanish) MEC Project MTM2012-38122-C03-02.

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Mundet i Riera, I. Finite group actions on 4-manifolds with nonzero Euler characteristic. Math. Z. 282, 25–42 (2016). https://doi.org/10.1007/s00209-015-1530-8

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