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

Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let R+(X) be the space of all such metrics. For any g is an element of R+(X), Hitchin used the KO-valued alpha-invariant to define a homomorphism A(n-1): pi(n-1)(R+(X), g) -> KOm+n. He then showed that A(0) not equal 0 if m = 8 k or 8k + 1 and that A(1) not equal 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) not equal 0 and pi(8j+1-m) (R+(X)) not equal 0 whenever m >= 7 and 8j - m >= 0. The new input are elements with nontrivial alpha-invariant deep down in the Gromoll filtration of the group Gamma(n+1) = pi(0)(Diff(D-n, partial derivative)). We show that alpha(Gamma(8j+2)(8j-5)) not equal {0} for j >= 1. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.