A simply connected numerical Campedelli surface with an involution

We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combine a double covering and $\mathbb{Q}$-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Koll\'ar and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced $\mathbb{Z}/2\mathbb{Z}$-action.