2016/12/31 by Diarmuid Crowley, Thomas Schick, Wolfgang Steimle · 1 citation
Mathematics · #math.GT #math.AT #math.DG
paper · pdf · doi:10.1112/topo.12081
published as Journal of Topology, Volume 11, Issue 4, 1077-1099 (2018) · Final version, to appear in Journal of Topology. 26 pages
arxiv created 2018/08/29 · arxiv updated 2018/11/22
We construct non-trivial elements of order 2 in the homotopy groups π8j+1+* Diff(D6,∂), for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in KO_*=Z/2. These elements are constructed by means of Morlet's homotopy equivalence between Diff(D6,∂) and Ω7(PL6/O6), and Toda brackets in PL6/O6. We also construct non-trivial elements of order 2 in π_* PLm for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in π_* Diff(Dm,∂) for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in π_* Diff(M) where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of (PL/O)(p) for the prime 2; (f) for any bP8j+4-sphere (where j > 0) of order which divides 4, the corresponding element in π0 Diff(D8j+2,∂) lifts to π8j-4 Diff(D6,∂), i.e., lies correspondingly deep down in the Gromoll filtration.