What kind of 3-manifolds can arise as hypersurfaces $\{ f(x,y,z,w) = 0\} \subset \mathbb{R}^4$? Can they have nontrivial H1 or H2?



What kind of 3-manifolds arise has hypersurfaces in R^4?
A simple construction that bears on the narrow version of John's question: If $M$ is a closed $n$-manifold that embeds in $\mathbb{R}^{n+1}$ (which can only happen if $M$ is orientable), then $M \times S^k$ embeds in $\mathbb{R}^{n+1+k}$. Thicken $M$ in $\mathbb{R}^{n+1}$, then cross with $I^k$ in the new dimensions, and then take the boundary of that. By induction, then, any product of spheres embeds in the next dimension.


On the other hand, Ryan Budney in [arXiv:0810.2346](http://arxiv.org/abs/0810.2346) has both new results and a bibliography of the broad question of which closed 3-manifolds embed in $\mathbb{R}^4$. It is problem 3.20 in Kirby's problem list, it is an interesting open problem, and there have been several partial results.


