Mathematics > Geometric Topology
arXiv:2603.18661 (math)
[Submitted on 19 Mar 2026 (v1), last revised 29 May 2026 (this version, v2)]
Abstract: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 $\lambda$-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 $\mu$-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.
| Comments: | 40 pages. This is an extended version of the previous version involving the spin counterparts. Also the proof of Proposition 3.1 is simplified. Comments are welcome! |
| Subjects: | Geometric Topology (math.GT); Algebraic Topology (math.AT) |
| MSC classes: | 57R19, 57R20, 57R55, 57R67, 11E81 |
| Cite as: | arXiv:2603.18661 [math.GT] |
| (or arXiv:2603.18661v2 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2603.18661 arXiv-issued DOI via DataCite |
Submission history
From: Fupeng Xu [view email]
[v1]
Thu, 19 Mar 2026 09:25:02 UTC (34 KB)
[v2]
Fri, 29 May 2026 05:24:28 UTC (54 KB)
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