Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups
Abstract
Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2018
- DOI:
- 10.48550/arXiv.1802.01757
- arXiv:
- arXiv:1802.01757
- Bibcode:
- 2018arXiv180201757Y
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Algebraic Topology;
- Mathematics - Dynamical Systems
- E-Print:
- Final version, to appear in the Quarterly Journal of Mathematics