/sci/ - Science & Math

>>12000877
>>12000892
>>12000992
the thing about covering spaces is the path lifting property. let [math]p \colon \tilde{X} \to X[/math] be a covering and let [math]\gamma \colon I \to X[/math] be a path with [math]\gamma(0) = x[/math]. then there exists a lift [math]\tilde{\gamma} \colon I \to \tilde{X}[/math] and this lift is unique up to the choice of basepoint [math]\tilde{\gamma}(0) \in p^{-1}(x)[/math]. see pic related. the important fact is that if [math]\gamma[/math] is a loop, the lift is not a loop in general, it can connect different points in the same fibre. this correspondence between {loops in [math]X[/math]} and {paths connecting points in the same fibre in [math]\tilde{X}[/math]} depends only on homotopy classes and it makes covering spaces essential in the study of the fundamental group. there are a lot of things to be said, I'll just say how it goes in the particular case. however I suggest you study this at some point, it's really a very nice part of algebraic topology.

let's now look at [math]Spin(n)[/math] and [math]SO(n)[/math]. the fibre of the identity element [math]1 \in SO(n)[/math] contains the identity element [math]\tilde{1}[/math] of [math]Spin(n)[/math] and also one another element, let's call it [math]-\tilde{1}[/math]. consider now a loop [math]\gamma[/math] in [math]SO(n)[/math] based at [math]1[/math], and lift it to a path [math]\tilde{\gamma}[/math] starting at [math]\tilde{1}[/math]. there are two possibilities: the end point of [math]\tilde{\gamma}[/math] is either [math]\tilde{1}[/math], or [math]-\tilde{1}[/math]. this depends exactly on whether [math]\gamma[/math] represents the trivial or the non-trivial class in [math]\pi_1(SO(n)) = \mathbb{Z}_2[/math]. thus the lifting procedure gives a bijection [math]\pi_1( SO(n) ) \cong \{ \tilde{1},-\tilde{1}\}[/math].