boundary of a simply connected, compact manifold with boundary For a simply connected, compact n-manifold with boundary (n > 1), is its boundary connected?
It’s obviously false when n = 1, but how to prove or disprove the statement when n > 1? I’m especially interested in n = 2. Thanks.

Let $X$ be a simply connected and compact $2$-manifold, hence a surface. According to "Riemann Surfaces" by Simon Donaldson (See here, Chapter 7, page 92), we have the following inequality for surfaces:
$$b_0(\partial X)\leq 2-\chi(X),$$
where $b_0(\partial X)$ (zeroth Betti number) denotes the number of path-connected components of $\partial X$. We need to compute those for $X$ to compute its Euler characteristic:

*

*Since $X$ is simply connected and therefore per definition path-connected, we have $b_0(X)=1$.

*Since $X$ is simply connected, we have $H_1^\text{sing}(X,\mathbb{Z})\cong\pi_1(X)^\mathrm{ab}\cong 1$ and therefore $b_1(X)=0$.

*Since the Betti number are the ranks of their respective homology groups, we have $b_2(X)\geq 0$.

*Since $X$ is a $2$-manifold, we have $b_n(X)=0$ for $n>2$.

We therefore have:
$$\chi(X)=b_0(X)-b_1(X)+b_2(X)\geq 1$$
resulting in:
$$b_0(\partial X)\leq 1,$$
which proves that $\partial X$ is path-connected and therefore connected.
