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

Manuel Krannich; Alexander Kupers; Oscar Randal-Williams

doi:10.5802/crmath.156

Differential topology

{HP}^{2}

-bundle over

S^{4}

with nontrivial Â-genus

Manuel Krannich ¹ ; Alexander Kupers ²; Oscar Randal-Williams ¹

¹ Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
² Department of Computer and Mathematical Sciences, University of Toronto Scarborough, 1265 Military Trail, Toronto, ON M1C 1A4, Canada

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 149-154.

Abstract
Résumé

We explain the existence of a smooth $H P^{2}$ -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.

Nous expliquons l’existence d’un fibré différentiel de base $S^{4}$ et fibre $H P^{2}$ , dont l’espace total est de $\hat{A}$ -genre non-trivial. En combinant ce resultat avec un argument de Hitchin, ceci répond à une question de Schick et implique que l’espace de métriques riemanniennes de courbure sectionnelle positive sur une variété fermée peut avoir des groupes d’homotopie rationnelle supérieures non-triviaux.

Received: 2020-07-31
Revised: 2020-11-23
Accepted: 2020-11-23
Published online: 2021-03-17

DOI: 10.5802/crmath.156

Classification: 57R20, 55R40, 57R22, 58D17

Author's affiliations:

Manuel Krannich ¹; Alexander Kupers ²; Oscar Randal-Williams ¹

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Manuel Krannich and Alexander Kupers and Oscar Randal-Williams},
     title = {An $\protect \text{HP}^2$-bundle over $\protect \text{S}^4$ with nontrivial {\^A-genus}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {149--154},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.156},
     language = {en},
}

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Manuel Krannich; Alexander Kupers; Oscar Randal-Williams. An $\protect \text{HP}^2$-bundle over $\protect \text{S}^4$ with nontrivial Â-genus. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 149-154. doi : 10.5802/crmath.156. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.156/

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