Abstract:Given a smooth, oriented, simply-connected $4$-manifold $M$, the homological Nielsen realization problem asks: when does a finite group of isometries $G\leq O(H_2(M;\mathbb{Z}))$ preserving the intersection form lift isomorphically to a finite group of orientation-preserving diffeomorphisms? We study this question for the smooth, positive-definite 4-manifolds $M_n:=\#_n\mathbb{CP}^2$. Even though every isometry of $H_2(M_n;\mathbb{Z})$ is induced by some orientation-preserving diffeomorphism, not necessarily of finite order, we show that Nielsen realization is sparse: as $n\to\infty$, a random subgroup of $O(H_2(M_n;\mathbb{Z}))$ is asymptotically almost never realizable in $\mathrm{Diff}^+(M_n)$; the same is true for random odd order elements of $O(H_2(M_n;\mathbb{Z}))$. We present both positive realization results in certain cases and a range of obstructions to realization in other cases. The proofs combine equivariant connected-sum constructions, fixed-point theory for group actions on 4-manifolds, finite group actions on surfaces, analytic combinatorics, and previous work of Hambleton--Tanase.
From: Ethan Pesikoff [view email]
[v1]
Tue, 26 May 2026 18:10:11 UTC (4,279 KB)
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