Topology of positively curved 8-dimensional manifolds with symmetry

TL;DR

This paper classifies 8-dimensional positively curved manifolds with certain symmetries, showing they closely resemble rank one symmetric spaces and providing strong classification results especially for torsion-free cases.

Contribution

It establishes that such manifolds share key topological features with classical symmetric spaces and offers a detailed classification for torsion-free cases.

Findings

Euler characteristic matches that of S^8, HP^2, or CP^4

Rationally elliptic manifolds are rationally isomorphic to rank one symmetric spaces

Torsion-free manifolds are classified more strongly

Abstract

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic then M is rationally isomorphic to a rank one symmetric space. For torsion-free manifolds we derive a much stronger classification. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank $\geq 2$. As an illustration we apply our results to various families of 8-manifolds.