Homology Sphere Embedding into $\mathbb R^4$

Question

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?

Answer 1

There are certainly closed hyperbolic 3-manifolds that embed in $\mathbb R^4$, that also have arbitrarily big volume. A construction goes as follows: construct a Mazur manifold with one 1-handle and one 2-handle. More specifically, by attaching a 1-handle to $D^4$ you get $D^3 \times S^1$. Then you attach a 2-handle along a knot $K \subset \partial (D^3 \times S^1) = S^2 \times S^1$ that is homotopic to the knot $\{ pt\} \times S^1$. The result is a contractible 4-manifold $W$ whose double is $S^4$ (see the Wikipedia page). Therefore $W$ embeds in $\mathbb R^4$ and hence also $\partial W$ does (smoothly). Moreover $\partial W$ is a homology sphere because it is the boundary of a contractible 4-manifold.

Now you have a lot of freedom here: you can take $K$ sufficiently complicated so that its complement $M = S^2\times S^1 \setminus K$ is hyperbolic and has arbitrarily big volume. You can attach the 2-handles along integral Dehn filling parameters that are arbitrarily big, so that the resulting manifold $\partial W$, which is in fact obtained by integral surgery from $K$, will still be hyperbolic with volume close to that of $M$ thanks to Thurston's Hyperbolic Dehn filling Theorem.

Answer 2

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