Gromov’s non-compact domination conjecture for positive scalar curvature

Establish that if a compact orientable manifold cannot be dominated by compact manifolds with positive scalar curvature, then it cannot be dominated by complete manifolds with positive scalar curvature, where domination denotes the existence of a map of nonzero degree (in particular, degree ±1 as considered in this context).

Background

The paper discusses how topological obstructions to positive scalar curvature (PSC) on closed manifolds extend to non-compact settings and highlights the role of domination maps (degree ±1 maps) in preserving such obstructions. Within this framework, Gromov’s conjecture proposes a non-compact analogue: PSC obstructions should persist under domination by complete manifolds when they already prevent domination by compact PSC manifolds.

The authors note that examples such as closed enlargeable manifolds and closed Schoen–Yau–Schick (SYS) manifolds cannot be dominated by manifolds with complete PSC metrics, pointing to the broader significance of the conjecture in understanding PSC obstructions beyond the compact category. Their work contributes to the compact-to-noncompact paradigm in SYS manifolds, aligning with the conjecture’s theme of invariance under domination.