Highly connected manifolds of positive $p$-curvature

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 in earlier papers. It turns out that the positivity of $p$-curvature is preserved under surgeries of codimension at least $p+3$. This gives a key to reducing 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 bordism groups $\Omega ^{\operatorname {spin}}_*$, $\Omega ^{\textup {string}}_*$, and by means of an $\alpha$-invariant and a Witten genus $\phi _W$. Here we use results from Anand Dessai (2009), which provide appropriate generators of the ring $\Omega ^{\textup {string}}_*\otimes \mathbf {Q}$ in terms of ``geometric ${\mathbb{C}}{\mathbf a}\mathbf {P}^2$-bundles'', where the Cayley projective plane ${\mathbb{C}}{\mathbf a} \mathbf {P}^2$ is a fiber and the structure group is $F_4$ which is the isometry group of the standard metric on ${\mathbb{C}}{\mathbf a}\mathbf {P}^2$.