The Gromoll filtration, KO–characteristic classes and metrics of positive scalar curvature

Crowley, Diarmuid¹ ; Schick, Thomas²

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
² Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstrasse 3, D-37073 Göttingen, Germany

Geometry & topology, Tome 17 (2013) no. 3, pp. 1773-1789.

Résumé

Let $X$ be a closed $m$–dimensional spin manifold which admits a metric of positive scalar curvature and let ${\mathcal{ℛ}}^{+}\left(X\right)$ be the space of all such metrics. For any $g\in {\mathcal{ℛ}}^{+}\left(X\right)$, Hitchin used the $\mathit{KO}$–valued $\alpha$–invariant to define a homomorphism $A_{n-1}:{\pi }_{n-1}\left({\mathcal{ℛ}}^{+}\left(X\right),g\right)\to {\mathit{KO}}_{m+n}$. He then showed that $A_0\ne 0$ if $m=8k$ or $8k+1$ and that $A_1\ne 0$ if $m=8k-1$ or $8k$.

In this paper we use Hitchin’s methods and extend these results by proving that

whenever $m\ge 7$ and $8j-m\ge 0$. The new input are elements with nontrivial $\alpha$–invariant deep down in the Gromoll filtration of the group ${\Gamma }^{n+1}={\pi }_0\left(Diff\left(D^n,\partial \right)\right)$. We show that $\alpha \left({\Gamma }_{8j-5}^{8j+2}\right)\ne \left\{0\right\}$ for $j\ge 1$. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

Détail

DOI : 10.2140/gt.2013.17.1773

Classification : 57R60, 53C21, 53C27, 58B20
Keywords: positive scalar curvature, $\alpha$–invariant, Gromoll filtration, exotic sphere

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