Euler characteristic = 1 iff manifod simply connected?

Question

let $D$ be a smooth manifold with boundary. Is it true that the Euler characteristic of $D$ is equal to 1 if and only if $D$ is simply connected? For domains in $\mathbb{R}^2$ this should be true, but I am not sure for general manifolds.

Edit: The answer below explains that this is false in general. But it is true for 2-dimensional manifolds with non-empty boundary.

Does anyone know a good source which explains such characteristics of the Euler characteristic?

Best wishes

Answer 1

This is false in general. Removing $k$ open balls from $S^3$ leaves a manifold of Euler characteristic $-k$, and that manifold is simply connected because it is homotopy equivalent to removing $k-1$ points from $\mathbb{R}^3$.