$\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913431507959808 |
|---|---|
| author | He, Siqi Parker, Gregory J. |
| author_facet | He, Siqi Parker, Gregory J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds He, Siqi Parker, Gregory J. Differential Geometry Analysis of PDEs Geometric Topology 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. |
| title | $\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds |
| topic | Differential Geometry Analysis of PDEs Geometric Topology |
| url | https://arxiv.org/abs/2407.10922 |