Infinite loop spaces and positive scalar curvature
Abstract
We study the homotopy type of the space of metrics of positive scalar curvature on highdimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$orientation from the infinite loop space of the MadsenTillmannWeiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the GromovLawson surgery technique which allows us to apply homotopytheoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other indextheoretical results, such as the AtiyahSinger family index theorem, the additivity theorem for indices on noncompact manifolds and the spectralflow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the highdimensional analogue of the MadsenWeiss theorem, proven by Galatius and the third named author.
 Publication:

Inventiones Mathematicae
 Pub Date:
 September 2017
 DOI:
 10.1007/s0022201707193
 arXiv:
 arXiv:1411.7408
 Bibcode:
 2017InMat.209..749B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 Mathematics  Geometric Topology;
 19D06;
 19K56;
 53C27;
 55P47;
 55R35;
 55S35;
 57R22;
 57R65;
 57R90;
 58D17;
 58D05;
 58J20
 EPrint:
 Final version, to appear in Inventiones Mathematicae