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Comptes Rendus. Mathématique
Differential topology
An HP 2 -bundle over S 4 with nontrivial Â-genus
Manuel Krannich1 ; Alexander Kupers2; Oscar Randal-Williams1
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.

We explain the existence of a smooth HP 2 -bundle over S 4 whose total space has nontrivial 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 HP 2 , dont l’espace total est de 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.156
Classification: 57R20,  55R40,  57R22,  58D17
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 
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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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