On the topology and geometry of certain $13$-manifolds

TL;DR

The paper classifies simply connected closed 13‑manifolds $\mathcal{M}$ with $H^*(\mathcal{M})\cong H^*(\mathrm{CP}^3\times S^7;\mathbb{Z})$ up to diffeomorphism, homeomorphism, and homotopy equivalence by exploiting normal $5$‑smoothing theory and surgery; the classification is governed by the first Pontrjagin class $p_1(\mathcal{M})$ and the group of exotic 13‑spheres $\Sigma^{13}$, with concrete statements linking $p_1$ to diffeomorphism types via connected sums with $\Sigma^{13}$ and to homeomorphism/homotopy types modulo small integers. The work develops the framework of bordism groups $\Omega_{13}^{O\langle8\rangle}(\xi)$, computes their structure using AHSS/ASS analyses, and shows how sphere bundles over $\mathrm{CP}^3$ realize all possible $p_1$‑values, enabling corollaries about diffeomorphism types and curvature. It also proves that total spaces of certain eight‑dimensional vector bundles over $\mathrm{CP}^3$ admit metrics of non‑negative sectional curvature, using explicit bundle constructions $\eta_{\boldsymbol{q}}$ whose $p_1$ can realize any integer, thereby connecting topology with Riemannian geometry. Overall, the paper delivers a precise topological classification for this family of 13‑manifolds and demonstrates curvature realizations for a broad class of associated bundles.

Abstract

This paper gives the classifications of certain manifolds $\mathcal{M}$ of dimension $13$ up to diffeomorphism, homeomorphism, and homotopy equivalence, whose cohomology rings are isomorphic to $H^\ast(\mathrm{CP}^3\times S^7;\mathbb{Z})$. Moreover, we prove that either $\mathcal{M}$ or $\mathcal{M}\#Σ^{13}$ admits a metric of non-negative sectional curvature where $Σ^{13}$ is a certain exotic sphere of dimension 13.

Figures (2)

• Figure 1: $\mathrm{Ext}^{s,t}_{\mathscr{A}_2}(\mathbb{Z}_2, \mathbb{Z}_2)$
• Figure 2: $E_1^{s,m,n}$ ($t=m+n$)

Theorems & Definitions (8)

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