#### Volume 17, issue 3 (2013)

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The Gromoll filtration, $\mathit{KO}$–characteristic classes and metrics of positive scalar curvature

### Diarmuid Crowley and Thomas Schick

Geometry & Topology 17 (2013) 1773–1789
##### Abstract

Let $X$ be a closed $m$–dimensional spin manifold which admits a metric of positive scalar curvature and let ${\mathsc{ℛ}}^{+}\left(X\right)$ be the space of all such metrics. For any $g\in {\mathsc{ℛ}}^{+}\left(X\right)$, Hitchin used the $\mathit{KO}$–valued $\alpha$–invariant to define a homomorphism ${A}_{n-1}:{\pi }_{n-1}\left({\mathsc{ℛ}}^{+}\left(X\right),g\right)\to {\mathit{KO}}_{m+n}$. 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

${A}_{8j+1-m}\ne 0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{\pi }_{8j+1-m}\left({\mathsc{ℛ}}^{+}\left(X\right)\right)\ne 0$

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.

##### Keywords
positive scalar curvature, $\alpha$–invariant, Gromoll filtration, exotic sphere
##### Mathematical Subject Classification 2010
Primary: 57R60
Secondary: 53C21, 53C27, 58B20