Extended fundamental-class obstruction to positive scalar curvature on compact manifolds

Determine whether, for every closed connected oriented smooth manifold M with fundamental group Γ and classifying map ν: M → BΓ, the nonvanishing of the pushforward ν∗[M] ∈ Hn(BΓ;Q) always obstructs the existence of a Riemannian metric of positive scalar curvature on M.

Background

This extension of the aspherical-case conjecture proposes that the image of the fundamental class under the classifying map provides a homological obstruction to positive scalar curvature on any compact manifold, not only aspherical ones.

The authors reference extensive work using Dirac operators to attack this extended conjecture, emphasizing its centrality in the paper of scalar curvature obstructions.