TITLE: Homology Sphere Embedding into $\mathbb R^4$
QUESTION [6 upvotes]: Let $Y$ be an oriented closed $3$-manifold, with trivial homology group, i.e. integer homological sphere. 
Q: If $Y$ can be embedded into $\mathbb R^4$, is there any example, that such a $Y$ admits a negative scalar curvature?

REPLY [6 votes]: L Z Gao and S T Yau showed in Invent Math 85 (1986) 637-652  that any compact 3 manifold admits a metric of negative Ricci curvature .J Lohkamp proved that any manifold of dimension 3 or higher admits a complete metric with Ricci curvature pinched between two negative constants .See Annals of Math 140 (1994) 653-683