# Example of a 3-dimensional manifold with toroidal boundary, which is not a solid torus

I am looking for an example of a $$3$$-dimensional compact and orientable topological manifold $$\mathcal{M}$$ with boundary, such that $$\partial\mathcal{M}$$ is homeomorphic to a $$2$$-torus $$T^{2}=S^{1}\times S^{1}$$, but such that $$\mathcal{M}$$ is not homeomorphic to the solid torus $$\overline{T}^{2}:=S^{1}\times\overline{D}$$, where $$\overline{D}$$ denotes the closed disk (=closed $$2$$-ball).

This is basically a similar question as in this post, which I've created some days ago. In this post I asked for manifolds, with spherical boundary, which are not homeomorphic to a $$3$$-ball. As explained by Moishe Kohan in this post, a general procedure to built them is to consider an arbitrary closed orientable $$3$$-manifold (which is not equivalent to $$S^{3}$$), from which we "cut out" the interior of a regular $$3$$-ball. Is there a similar procedure which works for the torus case?

I already know from the well-known theorem $$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M})$$ for odd-dimensional compact manifolds that every manifold $$\mathcal{M}$$ with toroidal boundary has to have $$\chi(\mathcal{M})=0$$. So, whenever someone knows an example of a $$3$$-dimensional compact and orientable manifold with a single boundary component, whose Euler-characterisitc is zero, then we would be done.

• Take a solid torus and connect sum with a closed 3 manifold that isn't a sphere. Commented Nov 9, 2021 at 18:29
• @ConnorMalin thanks for the comment, I think this is what I am looking for. I am only familiar with the connected sum of two closed manifold. Could you elaborate a little bit how the connected sum of a closed manifold with a manifold with boundary is defined? Commented Nov 9, 2021 at 19:28

In this situation there is no need to rule out $$S^3$$. In fact $$S^3$$ gives us gobs of beautiful examples closely related to knot complements: in fact what this construction shows is that every knot complement is the interior of a 3-manifold with torus boundary.
• I'd just like to add that a sufficient (but not necessary) condition is that, if after removing the solid torus, the resulting manifold $M$ is such that the induced map $\pi_1(\partial M)\to\pi_1(M)$ is injective. In this case, $M$ is definitely not a solid torus (this follows from the loop theorem). If this map is not injective, then $M$ is actually a connect sum with a solid torus. Commented Nov 10, 2021 at 20:04