AnHP2-bundle overS4with nontrivial Â-genus

Manuel Krannich,Alexander Kupers,Oscar Randal-Williams

doi:10.5802/crmath.156

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.

Highlights

In view of applications to spaces of Riemannian metrics with positive curvature, there has been recent interest in constructing smooth fibre bundles over spheres whose total space has nontrivial A-genus
In their work on the space of positive scalar curvature metrics, Hanke– Schick–Steimle [7, Corollary 1.6] showed that such bundles exist for every dimension of the base sphere
As noted in [7, p. 337], their method does not yield bundles with an explicit description of the fibre, though they are able to show that the fibre may be chosen to carry a metric of positive scalar curvature using a theorem of Stolz

Summary

Introduction

In view of applications to spaces of Riemannian metrics with positive curvature, there has been recent interest in constructing smooth fibre bundles over spheres whose total space has nontrivial A-genus. This factorisation follows for instance from [5, Theorem 1], but there is a more direct argument: via the canonical isomorphism π4(BDiff(HP 2)) ∼= π3(Diff(HP 2); id), the morphism A : π4(BDiff(HP 2) → Q is given by mapping a diffeomorphism φ : D3 × HP 2 → D3 × HP 2 that is the identity on the boundary and commutes with the projection to D3 to the A-genus of the glued manifold D4 × HP 2 ∪φ∪id D4 × HP 2.

Results

Conclusion