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

Abstract

We completely classify simply connected non-spin $7$-manifolds with only non-trivial middle homology groups $H_{2}\cong H_{4}\cong \mathbb{Z}\big/2$. They are referred to as $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds, and they have the minimal topological complexity among simply connected non-spin rational homology $7$-spheres that are not $2$-connected. We show that Milnor's $λ$-invariant establishes a bijection from oriented diffeomorphism classes of $\mathcal{G}_{3}(\mathrm{Wu})$-like manifolds to $\mathbb{Z}\big/7$, and each $\mathcal{G}_{3}(\mathrm{Wu})$-like manifold can be written as the connected sum of a standard $\mathcal{G}_{3}(\mathrm{Wu})$-like manifold and certain homotopy $7$-sphere.

Figures (13)

• Figure 1.1: The gyration $\mathcal{G}_{3}(\mathrm{Wu})$
• Figure 2.1: The normal $2$-type and normal $2$-smoothing of $\mathcal{G}_{3}(\mathrm{Wu})$-like manifold $M$
• Figure 3.1: The long exact ladder of Hurewicz homomorphisms associated to $\left(\mathcal{B}, W\right)$
• Figure 3.2: The long exact ladder of Hurewicz homomorphisms associated to $\left(\mathcal{B}, W\right)$ with groups identified
• Figure 3.3: The long exact braid of relative homology groups associated to $\left(\mathcal{B}, W, \partial W, M_{i}\right)$

Theorems & Definitions (42)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Proof 1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proof 2
• Proof 3: of Lemma \ref{['Lemma: normal 2-type of G_3(Wu)-like mfds']}