Geometry & Topology

Infinite loop spaces and positive scalar curvature in the presence of a fundamental group

Johannes Ebert and Oscar Randal-Williams

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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. This gives the first example of the nontriviality of the group C –algebra-valued secondary index invariant on higher homotopy groups. As an application, we produce a compact Spin 6 –manifold whose 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 manifolds with positive scalar curvature metrics.

Article information

Geom. Topol., Volume 23, Number 3 (2019), 1549-1610.

Received: 21 February 2018
Revised: 30 August 2018
Accepted: 7 October 2018
First available in Project Euclid: 5 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 19K56: Index theory [See also 58J20, 58J22] 53C27: Spin and Spin$^c$ geometry 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]

positive scalar curvature Gromov–Lawson surgery cobordism categories diffeomorphism groups Madsen–Weiss-type theorems secondary index invariant Rosenberg index Baum–Connes conjecture


Ebert, Johannes; Randal-Williams, Oscar. Infinite loop spaces and positive scalar curvature in the presence of a fundamental group. Geom. Topol. 23 (2019), no. 3, 1549--1610. doi:10.2140/gt.2019.23.1549.

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