On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914162796396544 |
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| author | Xu, Fupeng |
| author_facet | Xu, Fupeng |
| contents | We derive the $s$-invariants of certain simply connected $7$-manifolds whose second homology groups are isomorphic to $\mathbb{Z}^{2}$. We apply the $s$-invariants to give a partial classification of simply connected total spaces of circle bundles over $\left(\mathbb{C}P^{1}\times\mathbb{C}P^{2}\right)\#\mathbb{C}P^{3}$ up to diffeomorphism. As an application, we show that there is a simply connected $7$-manifold whose space and moduli space of positive Ricci curvature metrics both have infinitely many path components. We also determine bordism groups $Ω_{8}^{Spin}\left(K_{2}\right)$ and $Ω_{8}^{Spin}\left(K_{2};\mathrm{pr}_{1}^{*}γ^{1}\right)$ that are required in the deduction of $s$-invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics Xu, Fupeng Differential Geometry Algebraic Topology Geometric Topology 32Q10, 55R25, 55R40, 57R19, 57R20, 58D27 We derive the $s$-invariants of certain simply connected $7$-manifolds whose second homology groups are isomorphic to $\mathbb{Z}^{2}$. We apply the $s$-invariants to give a partial classification of simply connected total spaces of circle bundles over $\left(\mathbb{C}P^{1}\times\mathbb{C}P^{2}\right)\#\mathbb{C}P^{3}$ up to diffeomorphism. As an application, we show that there is a simply connected $7$-manifold whose space and moduli space of positive Ricci curvature metrics both have infinitely many path components. We also determine bordism groups $Ω_{8}^{Spin}\left(K_{2}\right)$ and $Ω_{8}^{Spin}\left(K_{2};\mathrm{pr}_{1}^{*}γ^{1}\right)$ that are required in the deduction of $s$-invariants. |
| title | On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics |
| topic | Differential Geometry Algebraic Topology Geometric Topology 32Q10, 55R25, 55R40, 57R19, 57R20, 58D27 |
| url | https://arxiv.org/abs/2511.08618 |