A remark on $\Lambda^2$-enlargeable manifolds

TL;DR

This paper investigates a modified notion of $ ext{Lambda}^2$-enlargeability for manifolds, showing they cannot admit positive scalar curvature metrics and providing an alternative proof for a related conjecture.

Contribution

It introduces a weaker condition for $ ext{Lambda}^2$-enlargeability and proves the same non-existence of positive scalar curvature, offering a new proof of Wang-Zhang's theorem.

Findings

Modified $ ext{Lambda}^2$-enlargeability still prevents positive scalar curvature.

Provides an alternative proof of Wang-Zhang's theorem.

Shows the importance of local conditions in enlargeability concepts.

Abstract

In this note, we consider the case where the condition ``constant near infinity" in the definition of $\Lambda^2$-enlargeable manifolds is replaced by the condition ``locally constant near infinity" and prove that a $\Lambda^2$-enlargeable manifold in this modified sense still cannot carry a complete Riemannian metric of positive scalar curvature. As a consequence, we give another proof of Wang-Zhang's theorem on the generalized Geroch conjecture for complete spin manifolds.