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

Krannich, Manuel; Kupers, Alexander; Randal-Williams, Oscar

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 $\mathbf{H}{P}^2$, dont l’espace total est de $\widehat{A}$-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 $\mathbf{H}{P}^2$-bundle over $S^4$ whose total space has nontrivial $\widehat{A}$-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.

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

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

[2] Burghelea, Dan The rational homotopy groups of $\mathrm{Diff}\left(M\right)$ and $\mathrm{Homeo}\left(M^n\right)$ in the stability range, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) (Lecture Notes in Mathematics), Volume 763, Springer (1979), pp. 604-626 | DOI | MR | Zbl

[3] Burghelea, Dan; Lashof, Richard 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 | DOI | MR | Zbl

[4] Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin 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 | Zbl

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

[6] Farrell, F. Thomas; Tuschmann, Wilderich 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) | DOI | MR | Zbl

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

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

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

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

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

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

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

[14] Wall, Charles T. C. 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 | Zbl