Mapping classes fixing an isotropic homology class of minimal genus 0 in rational 4-manifolds

Seraphina Eun Bi Lee

Vol. 339 (2025), No. 2, 283–308

DOI: 10.2140/pjm.2025.339.283

Abstract

For any $N\ge 1$, let $M_N$ denote the rational $4$-manifold ${ℂ\mathbb{P}}^2\#N\overline{{ℂ\mathbb{P}}^2}$. We study the stabilizer $\mathrm{Stab}<!--FUNCTION\; APPLICATION-->(w)$ of a primitive, isotropic class $w\in H_2(M_N;\mathbb{Z})$ of minimal genus $0$ under the natural action of the topological mapping class group $\mathrm{Mod}<!--FUNCTION\; APPLICATION-->(M_N)$ on $H_2(M_N;\mathbb{Z})$. Although most elements of $\mathrm{Stab}<!--FUNCTION\; APPLICATION-->(w)$ cannot be represented by homeomorphisms that preserve any Lefschetz fibration $M_N\to \mathrm{\Sigma }$, we show that every element of $\mathrm{Stab}<!--FUNCTION\; APPLICATION-->(w)$ can be represented by a diffeomorphism that almost preserves a holomorphic, genus-$0$ Lefschetz fibration $\mathrm{proj}<!--FUNCTION\; APPLICATION-->:M_N\to {ℂ\mathbb{P}}^1$ whose generic fibers represent the homology class $w$. We also answer the Nielsen realization problem for a certain maximal torsion-free, abelian subgroup ${\Lambda }_w$ of $\mathrm{Mod}<!--FUNCTION\; APPLICATION-->(M_N)$ by finding a lift of ${\Lambda }_w$ to ${\mathrm{Diff}<!--FUNCTION\; APPLICATION-->}^{+}(M_N)\le {\mathrm{Homeo}<!--FUNCTION\; APPLICATION-->}^{+}(M_N)$ under the quotient map $q:{\mathrm{Homeo}<!--FUNCTION\; APPLICATION-->}^{+}(M_N)\to \mathrm{Mod}<!--FUNCTION\; APPLICATION-->(M_N)$. This lift of ${\Lambda }_w$ can be made to almost preserve $\mathrm{proj}<!--FUNCTION\; APPLICATION-->:M_N\to {ℂ\mathbb{P}}^1$. All results of this paper also hold for every primitive, isotropic class $w\in H_2(M_N;\mathbb{Z})$ if $N\le 8$ because any such class has minimal genus $0$.

Keywords

rational surface, mapping class group, Nielsen realization problem, Lefschetz fibration

Mathematical Subject Classification

Primary: 57K40, 57S25

Milestones

Received: 6 July 2023

Revised: 10 September 2025

Accepted: 11 September 2025

Published: 21 September 2025