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<p>For a 3-manifold <i>M</i>, the twist group Twist(<i>M</i>) is the subgroup of the mapping class group Mod(<i>M</i>) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(<i>M</i>). We prove that a nontrivial subgroup <i>G</i> &lt; Twist(<i>M</i>) is realized by diffeomorphisms if and only if <i>G</i> is cyclic and <i>M</i> is a connected sum of lens spaces, including <i>S</i><sup>1</sup> × <i>S</i><sup>2</sup>. We also apply our methods to the Burnside problem for 3-manifolds and show that Diff(<i>M</i>) does not contain an infinite torsion group when <i>M</i> is reducible and not a connected sum of lens spaces.</p>

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Nielsen realization for sphere twists on 3-manifolds

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  • 发表日期：

    2025年9月29日

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  • Lei Chen,
  • Bena Tshishiku

摘要

For a 3-manifold M, the twist group Twist(M) is the subgroup of the mapping class group Mod(M) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(M). We prove that a nontrivial subgroup G < Twist(M) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces, including S1 × S2. We also apply our methods to the Burnside problem for 3-manifolds and show that Diff(M) does not contain an infinite torsion group when M is reducible and not a connected sum of lens spaces.