# Ramified cover of 3-ball

I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a link?

• link = a 1-dimensional submanifold with possibly nonempty boundary.

If answer is "YES", can we choose in addition the restriction of the covering at the boundary?

-
For the second part, the answer will be no in general. If one has a boundary which is a torus, and you require that the covering map be the elliptic involution, then the map won't extend in general over the 3-manifold, since this would extend to an involution on the manifold, which in general doesn't exist. –  Ian Agol Jan 30 '11 at 19:09
–  Ian Agol Jan 30 '11 at 19:31
@Agol, Thank you so much, your ref gives more than I wanted :) –  Anton Petrunin Jan 30 '11 at 20:14

Berstein and Edmonds prove in Cor. 6.3 that for an orientable 3-manifold $W$ with connected boundary, with a branched cover $\varphi: \partial W\to S^2$ of degree $n>3$, then there is a branched cover $\Phi: W\to D^3$ such that $\Phi_{|\partial W}=\varphi$. In another paper, Edmonds claims in Theorem 2.1 that Cor. 6.3 extends to maps $f: W\to D^3$ such that the boundary map is a branched cover of the same degree as $f$ (allowable). One can easily construct an allowable map $f:W\to D^3$ by mapping $\partial W$ to $S^2$ by a branched cover so that each component of $\partial W$ has positive degree $>2$ (with respect to the orientation induced by $W$), and extend to all of $W$ by coning off. Theorem 2.1 implies that this map is homotopic to a branched cover.
The hypothesis of degree $>2$ is necessary, since for example if one has a knot $K\subset S^3$ which is not (strongly) invertible, then $M=S^3-\mathcal{N}(K)$ is a manifold with torus boundary such that there is a degree 2 map $T^2=\partial M\to S^2$ which is the quotient of the elliptic involution, but which doesn't extend over $M$.