Infinite loop spaces and positive scalar curvature

Abstract

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional 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 ≥ 6, the natural KO-orientation from the infinite loop space of the Madsen-Tillmann-Weiss 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 ≥ 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 Gromov-Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah-Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow 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 high-dimensional analogue of the Madsen-Weiss theorem, proven by Galatius and the third named author.

On the topology of the space of Ricci-positive metrics

“…It relies on computations by the second-named author and Randal-Williams [10] (which by themselves rely on many deep ingredients from high-dimensional manifold theory, such as [19] and [18], among other things). 3 The transgression trg d is the map H n (X; Q) ∼ = [X; K(Q, n)] *…”

“…An obstruction theory argument (see e.g. [3,Lemma 3.29]) shows that these bundles admit spin structures (since the structure group fixes a disc in the fibre).…”

“…and it has been proven [3] that (1) is surjective after rationalization for each spin manifold of dimension d ≥ 6 that admits a psc metric. More recently, Perlmutter [15,16] showed how to cover the case d = 5.…”

Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets

“…Acknowledgements. We would like to thank Peter May for helpful comments concerning the splitting of (P L/O) (p) in (5). We also thank the referee for many suggestions which have improved the presentation.…”

Moduli spaces of manifolds: a user's guide

“…[Kup16] to establish the finite generation of homotopy groups of Diff ∂ (D d ) for d = 4, 5, 7. (ii) These results have been used by Botvinnik, Ebert, and Randal-Williams[BERW17], Ebert and Randal-Williams[ERW17], and Botvinnik, Ebert, and Wraith[BEW18] to study the topology of spaces of Riemannian metrics of positive scalar, or Ricci, curvature. (iii) These results have been used by Krannich[Kra18] to show that if Σ is a homo-…”