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

We compute the relative smooth surgery structure sets of the thickenings ${\mathbb{O}\mathbb{P}}^2\times {\mathbb{D}}^k$ of the Cayley projective plane ${\mathbb{O}\mathbb{P}}^2$ for every $k\ge 1$ with $k\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, 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 ${\mathbb{O}\mathbb{P}}^2\times {\mathbb{S}}^k$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\stackrel{}{\mathrm{Diff}}({\mathbb{O}\mathbb{P}}^2)$ in every degree congruent to $3$ modulo $4$; and we construct smooth ${\mathbb{O}\mathbb{P}}^2$-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 ${\mathbb{O}\mathbb{P}}^2$ in degrees $3$, $7$, and $11$.