Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929514468081664 |
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| author | Nguyen, Minh Lam |
| author_facet | Nguyen, Minh Lam |
| contents | In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $\mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an $\mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology $HM$. This $\mathbf{R}$-filtration fits $HM$ into a persistent homology theory, from which one can define a numerical quantity called the spectral invariant $ρ$. The spectral invariant $ρ$ is tied with the geometry of the underlying manifold. The main result of the papers shows that $ρ$ provides an obstruction to the existence of positive scalar curvature metric on a ribbon homology cobordism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres Nguyen, Minh Lam Geometric Topology Differential Geometry 57Rxx, 57Mxx, 57Kxx In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $\mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an $\mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology $HM$. This $\mathbf{R}$-filtration fits $HM$ into a persistent homology theory, from which one can define a numerical quantity called the spectral invariant $ρ$. The spectral invariant $ρ$ is tied with the geometry of the underlying manifold. The main result of the papers shows that $ρ$ provides an obstruction to the existence of positive scalar curvature metric on a ribbon homology cobordism. |
| title | Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres |
| topic | Geometric Topology Differential Geometry 57Rxx, 57Mxx, 57Kxx |
| url | https://arxiv.org/abs/2409.04954 |