Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset \widetilde{M}$ such that

1. the union of $\gamma D$ over all $\gamma\in \pi_1(M)$ covers $\widetilde{M}$,

2. the collection $\gamma D^o$ are mutually disjoint,

3. $\pi(D)=M$ and the restriction $\pi|_{D^o}:D^o\to M$ is diffeomorphic onto its image.

My question is:

Does there always exist some simply connected fundamental domain?

Is every fundamental domain simply connected?

Motivation.

I saw the following statement in several papers about dynamical systems: let $B^d(0,1)$ be the unit ball in $\mathbb{R}^d$ and $M$ be a $d$-dimensional compact connected manifold without boundary. Then $M\simeq B^d(0,1)/\sim$ where $\sim $ represents some gluing along $S^{d-1}=\partial B^d(0,1)$.

I think the statement might be related to above question.

Thanks!