Isotopy classes of involutions of del Pezzo surfaces

Abstract

Let $M_n:={\mathbb{CP}}^2\#n\overline{{\mathbb{CP}}^2}$ for $0\le n\le 8$ be the underlying smooth manifold of a degree $9-n$ del Pezzo surface. We prove three results about the mapping class group $\text{Mod}(M_n):={\pi }_0({\text{Homeo}}^{+}(M_n))$: 1. the classification of, and a structure theorem for, all involutions in $\text{Mod}(M_n)$, 2. a positive solution to the smooth Nielsen realization problem for involutions of $M_n$, and 3. a purely topological characterization of three remarkable types of involutions on certain $M_n$ coming from birational geometry: de Jonquiéres involutions, Geiser involutions, and Bertini involutions. One main ingredient is the theory of hyperbolic reflection groups.