The Gromoll filtration, KOcharacteristic classes and metrics of positive scalar curvature
Abstract
Let X be a closed mdimensional spin manifold which admits a metric of positive scalar curvature and let Pos(X) be the space of all such metrics. For any g in Pos(X), Hitchin used the KOvalued alphainvariant to define a homomorphism A_{n1} from \pi_{n1}(Pos(X) to KO_{m+n}. He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0 if m = 8k1 or 8$. In this paper we use Hitchin's methods and extend these results by proving that A_{8j+1m} is not 0 whenever m>6 and 8j  m >= 0. The new input are elements with nontrivial alphainvariant deep down in the Gromoll filtration of the group \Gamma^{n+1} = \pi_0(\Diff(D^n, \del)). We show that \alpha(\Gamma^{8j+2}_{8j5}) is not 0 for j>0. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.6474
 Bibcode:
 2012arXiv1204.6474C
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry
 EPrint:
 14 pages, amslatex. v2: corrections. Based on a referee's report we added Lemma 2.5 v2 and we gave more details in the proof of Lemma 2.14 v2. We also removed Corollary 1.2 v1 since we found a gap in the proof. v3: typo (wrong index) in Lemma 2.14 corrected