Z₂-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds

Given a pair of Z₂-harmonic spinors (resp. 1-forms) on closed Riemannian 3-manifolds (Y₁, g₁) and (Y₂,g₂), we construct Z₂-harmonic spinors (resp. 1-forms) on the connected sum Y₁ # Y₂ and the torus sum Y₁ ∪T² Y₂ using a gluing argument. The main tool in the proof is a parameterized version of the Nash-Moser implicit function theorem established by Donaldson and the second author. We use these results to construct an abundance of new examples of Z₂-harmonic spinors and 1-forms. In particular, we prove that for every closed 3-manifold Y, there exist infinitely many Z₂-harmonic spinors with singular sets representing infinitely many distinct isotopy classes of embedded links, strengthening an existence theorem of Doan-Walpuski. Moreover, combining this with previous results, our construction implies that if b₁(Y) > 0, there exist infinitely many spinᶜ structures on Y such that the moduli space of solutions to the two-spinor Seiberg-Witten equations is non-empty and non-compact.