Relative Smooth Surgery Structure Sets of Thickenings of the Cayley Projective Plane and Applications

Abstract

We compute the relative smooth surgery structure sets of the thickenings $\mathbb{OP}^{2}\times\mathbb{D}^{k}$ of the Cayley projective plane $\mathbb{OP}^{2}$ for every $k\geq 1$ with $k\equiv 0\pmod 4$, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the $2$-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension $16+k$, homotopy equivalent to $\mathbb{OP}^{2}\times\mathbb{S}^{k}$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})$ in every degree congruent to $3$ modulo $4$; and we construct smooth $\mathbb{OP}^{2}$-bundles over $\mathbb{S}^{4}$, $\mathbb{S}^{8}$, and $\mathbb{S}^{12}$ whose total spaces have non-vanishing $\widehat{\mathfrak{A}}$-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on $\mathbb{OP}^{2}$ in degrees $3$, $7$, and $11$.