Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$
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| Format: | Preprint |
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2026
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| _version_ | 1866911619445948416 |
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| author | Baraglia, David Hekmati, Pedram |
| author_facet | Baraglia, David Hekmati, Pedram |
| contents | We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsváth-Szabó $d$-invariants, or equivalently the Seiberg-Witten $δ$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$ Baraglia, David Hekmati, Pedram Geometric Topology Differential Geometry We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsváth-Szabó $d$-invariants, or equivalently the Seiberg-Witten $δ$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute. |
| title | Floer homotopy type and eta invariants of Seifert $3$-manifolds fibering over $\mathbb{RP}^2$ |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2604.19195 |