Invent. math. (2017) 209:749–835
Inﬁnite loop spaces and positive scalar curvature
· Johannes Ebert
Received: 2 March 2016 / Accepted: 12 January 2017 / Published online: 14 February 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We study thehomotopytype of the space of metrics of positivescalar
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
any2n-dimensionalspinmanifold. Formanifolds of odd dimension 2n+1 ≥ 7,
we prove the existence of a similar factorisation. When combined with compu-
tational methods from homotopy theory, these results have strong implications.
Johannes Ebert was partially supported by the SFB 878.
Oscar Randal-Williams acknowledges Herchel Smith Fellowship support.
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstr. 62,
48149 Münster, Germany
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WB, UK