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.

This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial, for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum-Connes conjecture. For example, we produce a compact Spin 6manifold such that its space of positive scalar curvature metrics has each rational homotopy group infinite dimensional.At a more technical level, we introduce the notion of "stable metrics" and prove a basic existence theorem for them, which generalises the Gromov-Lawson surgery technique, and we also give a method for rounding corners of manifold with positive scalar curvature metrics.

This is an expository article without any claim of originality. We give a complete and self-contained account of the Gromov–Lawson–Chernysh surgery theorem for positive scalar curvature metrics.

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Abstract. Given two metrics of positive scalar curvature on a closed spin manifold, there is a secondary index invariant in real K-theory. There exist two definitions of this invariant, one of homotopical flavor, the other one defined by an index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this result to the case of two families of positive scalar curvature metrics, parametrized by a finite CW complex. In essence, we prove a generalization of the classical "spectral-flow-index theorem" to the case of families of real operators.