What is the boundary of tubular neighborhood of a compact manifold?

Question

What is the boundary of tubular neighborhood of a compact manifold? for example what is the boundary of tubular neighborhood of a torus $\mathbb{T}^2$ in $\mathbb{R}^3$ or a torus $\mathbb{T}^2$ in $\mathbb{R}^4$.

Answer 1

If you have a compact submanifold $M^n\subset\Bbb R^N$, the boundary of a (sufficiently small) tubular neighborhood will be a $S^{N-n-1}$-bundle over $M$. In both examples you posed, because the normal bundle is trivial, the $S^0$- and $S^1$-bundles, respectively, are both diffeomorphic to global products. However, for example, if you take a Klein bottle in $\Bbb R^4$, the boundary of a small tubular neighborhood in this case will be a "twisted" circle bundle.

To get more interesting examples, you'll want to consider compact submanifolds $M$ of a more general manifold, rather than of Euclidean space.