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Comptes Rendus. Mathématique
Topologie différentielle
An HP 2 -bundle over S 4 with nontrivial Â-genus
Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 149-154.

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.

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.

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DOI : https://doi.org/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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     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},
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     volume = {359},
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     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, Tome 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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