On the simplest simply connected rational homology $7$-spheres that are not $2$-connected
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| Format: | Preprint |
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2026
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| _version_ | 1866916063189401600 |
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| author | Xu, Fupeng |
| author_facet | Xu, Fupeng |
| contents | We give a complete classification of two families of simply connected $7$-manifolds: $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds and $\mathcal{G}_{3}^{p}(S^{5})$-like manifolds for odd primes $p$. The former are non-spin with $H_{2}\cong H_{4}\cong \mathbb{Z}/2$ as their only nontrivial middle homology; the latter have $H_{2}\cong H_{4}\cong \mathbb{Z}/p$ as their sole nontrivial middle homology. These manifolds attain the minimal homological complexity among simply connected rational homology $7$-spheres that are not $2$-connected.
We prove that Milnor's $λ$-invariant gives a bijection from the oriented diffeomorphism classes of $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds onto $\mathbb{Z}/7$, and each such manifold decomposes as the connected sum of a standard $\mathcal{G}_{3}(\mathrm{Wu})$-like manifold and a homotopy $7$-sphere. Analogously, the Eells-Kuiper $μ$-invariant yields a bijection from the oriented diffeomorphism classes of $\mathcal{G}_{3}^{p}(S^{5})$-like manifolds to $\mathbb{Z}/28$, with every manifold splitting as the connected sum of a standard $\mathcal{G}_{3}^{p}(S^{5})$-like manifold and a homotopy $7$-sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the simplest simply connected rational homology $7$-spheres that are not $2$-connected Xu, Fupeng Geometric Topology Algebraic Topology 57R19, 57R20, 57R55, 57R67, 11E81 We give a complete classification of two families of simply connected $7$-manifolds: $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds and $\mathcal{G}_{3}^{p}(S^{5})$-like manifolds for odd primes $p$. The former are non-spin with $H_{2}\cong H_{4}\cong \mathbb{Z}/2$ as their only nontrivial middle homology; the latter have $H_{2}\cong H_{4}\cong \mathbb{Z}/p$ as their sole nontrivial middle homology. These manifolds attain the minimal homological complexity among simply connected rational homology $7$-spheres that are not $2$-connected. We prove that Milnor's $λ$-invariant gives a bijection from the oriented diffeomorphism classes of $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds onto $\mathbb{Z}/7$, and each such manifold decomposes as the connected sum of a standard $\mathcal{G}_{3}(\mathrm{Wu})$-like manifold and a homotopy $7$-sphere. Analogously, the Eells-Kuiper $μ$-invariant yields a bijection from the oriented diffeomorphism classes of $\mathcal{G}_{3}^{p}(S^{5})$-like manifolds to $\mathbb{Z}/28$, with every manifold splitting as the connected sum of a standard $\mathcal{G}_{3}^{p}(S^{5})$-like manifold and a homotopy $7$-sphere. |
| title | On the simplest simply connected rational homology $7$-spheres that are not $2$-connected |
| topic | Geometric Topology Algebraic Topology 57R19, 57R20, 57R55, 57R67, 11E81 |
| url | https://arxiv.org/abs/2603.18661 |