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

TL;DR

This paper constructs a smooth $HP^2$-bundle over $S^4$ with a total space exhibiting a nontrivial $\hat{A}$-genus, revealing new insights into the topology of manifolds with positive curvature.

Contribution

It demonstrates the existence of a specific bundle with nontrivial $\hat{A}$-genus, addressing a question about the topology of positively curved manifolds.

Findings

Existence of a smooth $HP^2$-bundle over $S^4$ with nontrivial $\hat{A}$-genus.

Implication that the space of positive sectional curvature metrics can have nontrivial higher rational homotopy groups.

Answers a question posed by Schick using an argument related to Hitchin.

Abstract

We explain the existence of a smooth $HP^2$-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.