${\mathbb{Z}}_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds

Abstract

Given a pair of ${\mathbb{Z}}_2$-harmonic spinors (resp. 1-forms) on closed Riemannian 3-manifolds $(Y_1,g_1)$ and $(Y_2,g_2)$, we construct ${\mathbb{Z}}_2$-harmonic spinors (resp. 1-forms) on the connected sum $Y_1\#Y_2$ and the torus sum $Y_1{\cup }_{T^2}{Y}_2$ 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 ${\mathbb{Z}}_2$-harmonic spinors and 1-forms. In particular, we prove that for every closed 3-manifold $Y$, there exist infinitely many ${\mathbb{Z}}_2$-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 $b_1(Y)>0$, there exist infinitely many ${\mathrm{spin}}^c$ structures on $Y$ such that the moduli space of solutions to the two-spinor Seiberg-Witten equations is non-empty and non-compact.

40 pages, minor revisions to original version