On the simplest simply connected rational homology $7$-spheres that are not $2$-connected
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.