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Comptes Rendus. Mathématique
Differential topology
An HP 2 -bundle over S 4 with nontrivial Â-genus
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/

[1] Boris Botvinnik; Johannes Ebert; David J. Wraith On the topology of the space of Ricci-positive metrics, Proc. Am. Math. Soc., Volume 148 (2020), pp. 3997-4006 | Article | MR: 4127843 | Zbl: 1444.53030

[2] Dan Burghelea The rational homotopy groups of Diff(M) and Homeo(M n ) in the stability range, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) (Lecture Notes in Mathematics), Volume 763 (1979), pp. 604-626 | Article | MR: 561241 | Zbl: 0421.58008

[3] Dan Burghelea; Richard Lashof Geometric transfer and the homotopy type of the automorphism groups of a manifold, Trans. Am. Math. Soc., Volume 269 (1982) no. 1, pp. 1-38 | Article | MR: 637027 | Zbl: 0489.57008

[4] Dan Burghelea; Richard Lashof; Melvin Rothenberg Groups of automorphisms of manifolds, Lecture Notes in Mathematics, 473, Springer, 1975, vii+156 pages with an appendix (“The topological category”) by E. Pedersen | MR: 0380841 | Zbl: 0307.57013

[5] Johannes Ebert; Oscar Randal-Williams Generalised Miller–Morita–Mumford classes for block bundles and topological bundles, Algebr. Geom. Topol., Volume 14 (2014) no. 2, pp. 1181-1204 | Article | MR: 3180831 | Zbl: 1291.57018

[6] F. Thomas Farrell; Wilderich Tuschmann Mini-workshop: Spaces and moduli spaces of Riemannian metrics, Oberwolfach Rep., Volume 14 (2017) no. 1, pp. 133-166 (abstracts from the mini-workshop held January 8–14, 2017) | Article | MR: 3757071 | Zbl: 1390.00089

[7] Bernhard Hanke; Thomas Schick; Wolfgang Steimle The space of metrics of positive scalar curvature, Publ. Math., Inst. Hautes Étud. Sci., Volume 120 (2014), pp. 335-367 | Article | MR: 3270591 | Zbl: 1321.58008

[8] Friedrich Hirzebruch Über die quaternionalen projektiven Räume, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., Volume 1953 (1954), pp. 301-312 | MR: 65155 | Zbl: 0057.15503

[9] Friedrich Hirzebruch Topological methods in algebraic geometry, Classics in Mathematics, Springer, 1995, xii+234 pages (reprint of the 1978 edition) | MR: 1335917 | Zbl: 0843.14009

[10] Nigel Hitchin Harmonic spinors, Adv. Math., Volume 14 (1974), pp. 1-55 | Article | MR: 358873 | Zbl: 0284.58016

[11] Kiyoshi Igusa The stability theorem for smooth pseudoisotopies, K-Theory, Volume 2 (1988) no. 1-2, pp. 1-355 | Article | MR: 972368 | Zbl: 0691.57011

[12] Oscar Randal-Williams An upper bound for the pseudoisotopy stable range, Math. Ann., Volume 368 (2017) no. 3-4, pp. 1081-1094 | Article | MR: 3673647 | Zbl: 1378.57042

[13] Thomas Schick The topology of positive scalar curvature, Proceedings of the International Congress of Mathematicians (Seoul 2014). Vol. II (2014), pp. 1285-1307 | MR: 3728662 | Zbl: 1373.53053

[14] Charles T. C. Wall Surgery on compact manifolds, Mathematical Surveys and Monographs, 69, American Mathematical Society, 1999, xvi+302 pages (edited and with a foreword by A. A. Ranicki) | MR: 1687388 | Zbl: 0935.57003

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