Simple question on orthocronous proper lorentz transformation

Question

I've found written in an undergrad textbook on relativity (Barone's, Relativity; italian book, don't actually know if it's been translated in any other language) that proper orthocronous Lorentz transformation (for simplicity sake we'll say SO(1,3) transformation) form a group which is (i think the proper way of saying would be "a Lie group the manifold associated to which is") twofold-connected (hope that's the right translation); what i don't understand is why twofold-connected and not just simply connected as i can intuitively understand.

Answer 1

The group SO(3,1) has two discrete transformations, changing the unit sign of the determinant, T, the diagonal matrix of time reflection $t\to- t$ and R, the diagonal matrix $x_3 \to -x_3$.

All other spatial reflections can by accomplished as products of rotations and the single spatial reflection R acting on a single spatial dimension.

It follows, that $SO(3,1)$ has four distinct copies of the proper Lorentz $L_{t+,\det=1}$ with positive time direction and standard orientation of the space sub-matrix.

$L$ is the subgroup of the group, continously connected to the unit matrix. The four distinct constituents are $$SO(3,1) = ( \text{Id} \vee T \vee R \vee R\cdot T ) \cdot SO(3,1)_{t+,det=1} $$

To come the question, the orthochronous, $\det 1$, subgroup is the product of the 1-d $(t,x)$ Lorentz boost subgroup and the subgroup of proper rotations in $SO(3)_{det=1}$.

This group may be parametrized by a-d plane of the rotation and an angle of rotation. The classical parametrization uses the axis in $\mathbb R^3 $ and the angle as radius.

Now, by the fact, that the orientation of the plane or the direction of the axis leads to identify any rotions by $\pm \pi,$ the parameter space is the ball for the axis directions and the radius as the angle of rotation.

With this model of the group manifold, it's evident, that a radial path from $r=0$ to $r=\pi$ enters the ball over the antipodal point of the same axis with closing the path until it reaches the origin.

Such paths passing over the two identified antipodal points cannot be deformed continously into a point. By rotational invariance, on has one homotopy class of non contractible, closed single loops.

SU(2,C), as a more elementary representation of the rotation group, extends the parameter ball up to radius $4\pi$, with the areas of the outer 2-spheres beyond radius $2 \pi$ geometrically shrinking to a point at $4\pi$ again, that is identified with the identity.

SU(2,C) is simply connected with the consequence, that it cannot not represent coordinate reflections, because $-1$ is the value on the $2\pi$-\sphere.