An $\protect \text{HP}^2$-bundle over $\protect \text{S}^4$ with nontrivial Â-genus

Abstract

We explain the existence of a smooth $\mathbf{H} P^2$-bundle over $S^4$ whose total space has nontrivial $\hat{A}$-genus. Combined with an argument going back to Hitchin, this answers a question of Schick and implies that the space of Riemannian metrics of positive sectional curvature on a closed manifold can have nontrivial higher rational homotopy groups.