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We explain the existence of a smooth $HP^2$-bundle over $S^4$ whose totalspace has nontrivial $\hat{A}$-genus. Combined with an argument going back toHitchin, this answers a question of Schick and implies that the space ofRiemannian metrics of positive sectional curvature on a closed manifold canhave nontrivial higher rational homotopy groups.

An $HP^2$-bundle over $S^4$ with nontrivial $\hat{A}$-genus