Disjoint union of the torus and the sphere is a boundary of a compact manifold.

Question

What is the disjoint union of the torus $S^1\times S^1$ and the 2-sphere $S^2$?

I ask, because I am trying to prove that the torus is cobordant to the sphere. By definition, this means that their disjoint union is the boundary of a compact maniofold with dimension 3.

So, my next question is: Is the disjoint union the boundary of a compact manifold of dimension 3?

Answer 1

The tangent bundle to the torus is trivial, so its Stiefel-Whitney classes vanish. While the tangent bundle to the sphere is stably trivial. The Stiefel-Whitney class is stable, so the sphere's Stiefel-Whitney classes also vanish. It is a theorem that there exists a bordism between two manifolds if and only if they have the same Stiefel-Whitney numbers. Therefore there is a bordism between the torus and the sphere.

More explicitly, let's take a large 3-ball, and remove a small open solid torus from the center. A ball with a ring hollowed out. Here we have a 3-manifold with boundary a 2-sphere union a torus.

Answer 2

Take a solid torus in $\mathbb R^3$ and remove a small open ball from the inside.