I am currently reading this paper: https://arxiv.org/abs/gr-qc/9512041 . It is investigating the spin structures connection to skein modules.
I'm having a hard time understanding a part of the proof of theorem 1. We have a bundle of orthonormal frames. The author describes the lifts of two diagrams (to this bundle) as being specific elements in $H_1(SO(3), \mathbb{Z}_2)$.
Of the diagram above, the author says: "Therefore the difference in their homology classes is just the homology class of the lift of $p_1$, which consists of oriented differentiable segments. The lift of $p_1$ is a continuous curve and its homology class is zero." Why is the lift of oriented differentiable segments to the bundle of orthonormal frames a continuous curve?
Of this diagram, the author says: "this time the homology class of $\tilde{p_2}$ is the generator of $H_1(SO(3), \mathbb{Z}_2)$."
I guess I'm just struggling to understand what the lifts of these diagrams look like in the tangent bundle. Also, the arrows on the diagrams are confusing me. Any insight would be appreciated.