On the existence of harmonic ${\mathbf{Z}}_2$ spinors

We prove the existence of singular harmonic ${\mathbf{Z}}_2$ spinors on $3$-manifolds with $b_1>1$. The proof relies on a wall-crossing formula for solutions to the Seiberg-Witten equation with two spinors. The existence of singular harmonic ${\mathbf{Z}}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce regarding Donaldson and Segal’s proposal for counting $G_2$-instantons.