Wikipedia chapter about topology of 3D rotation group says:
Consider the solid ball in $\mathbb {R} ^{3}$ of radius $\pi$ (that is, all points of $\mathbb{R} ^{3}$ of distance $\pi$ or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball.
followed by:
consider the path running from the "north pole" straight through the interior down to the south pole [...] this loop represents a continuous sequence of rotations about the z-axis starting and ending at the identity rotation (i.e. a series of rotation through an angle $\varphi$ where $\varphi$ runs from $0$ to $2\pi$).
I do not understand why "north pole" (and also south pole) is the identity rotation. As north pole has cartessian coordinates $(0,0,\pi)$, according to first paragraph it must be understood as a rotation around z axis by a rotation angle $\pi$, that is not the identity rotation.
Could be due to this misunderstood, I do not reach my true target, understand why rotations has a fundamental group of order $2$.
The north pole is a clockwise rotation with angle $\pi$ and the south pole is a counterclockwise rotation with angle $\pi$. But these are equal (since the angle is $\pi$; otherwise, they would be two distinct rotations). And, while you go from the north pole to the south pole through the center of the sphere, you pass throuh that center, which corresponds to the identity rotation.