An $HP^2$-bundle over $S^4$ with nontrivial $\hat{A}$-genus
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2007.15062
- arXiv:
- arXiv:2007.15062
- Bibcode:
- 2020arXiv200715062K
- Keywords:
-
- Mathematics - Algebraic Topology;
- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology;
- 57R20;
- 55R40;
- 57R22;
- 58D17
- E-Print:
- 5 pages, to appear in Comptes Rendus. Math\'ematique