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

TL;DR

This paper classifies a specific class of simply connected, non-spin 7-manifolds with minimal topological complexity, establishing a bijection with the cyclic group of order 7 via Milnor's lambda-invariant.

Contribution

It provides a complete classification of certain non-spin rational homology 7-spheres, linking their diffeomorphism classes to a cyclic group and describing their structure via connected sums.

Findings

Milnor's lambda-invariant bijects with Z/7

Classification of G_3(Wu)-like manifolds

Decomposition into connected sums with homotopy spheres

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 $\lambda$-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.