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

Abstract

For any $N\ge 1$, let $M_N$ denote the rational $4$-manifold ${\mathbb{CP}}^2\#N\overline{{\mathbb{CP}}^2}$. In this paper we study the stabilizer $\text{Stab}(w)$ 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 $\text{Mod}(M_N)$ on $H_2(M_N;\mathbb{Z})$. Although most elements of $\text{Stab}(w)$ cannot be represented by homeomorphisms that preserve any Lefschetz fibration $M_N\to \Sigma$, we show that any element of $\text{Stab}(w)$ can be represented by a diffeomorphism that almost preserves a holomorphic, genus-$0$ Lefschetz fibration $\text{pr}:M_N\to {\mathbb{CP}}^1$ 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 $\text{Mod}(M_N)$ by finding a lift of ${\Lambda }_w$ to ${\text{Diff}}^{+}(M_N)\le {\text{Homeo}}^{+}(M_N)$ under the quotient map $q:{\text{Homeo}}^{+}(M_N)\to \text{Mod}(M_N)$ which can be made to almost preserve $\text{pr}:M_N\to {\mathbb{CP}}^1$. 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 $0$.

References in corpus (1)

• Elements and cyclic subgroups of finite order of the Cremona group