Highly connected manifolds of positive $p$-curvature

TL;DR

This paper classifies highly connected manifolds with positive p-curvature, linking geometric conditions to topological invariants and bordism groups, especially focusing on 3-connected manifolds with positive 2-curvature.

Contribution

It establishes that positive p-curvature is preserved under certain surgeries and reduces the classification problem to topological bordism and index theory, providing explicit classifications for specific cases.

Findings

Positivity of p-curvature preserved under surgeries of codimension at least p+3

Classification of 3-connected manifolds with positive 2-curvature using bordism groups and invariants

Use of geometric $ ext{Ca} P^2$-bundles to generate string bordism ring

Abstract

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $\alpha$-invariant and Witten genus $\phi_W$. Here we use results of Dessai, which provide appropriate generators of the rational string bordism ring in terms of "geometric $\Ca P^2$-bundles", where the Cayley projective plane $\Ca P^2$ is a fiber and the structure group is $F_4$ which is the isometry group of the standard metric on $\Ca P^2$.