notes / Math 55b / chapter 4: The Fundamental Group and Covering Spaces / From my notebook
N21 Classifying Covering Spaces
In progress: these notes stop partway.
Lecture 17
Recall: classifying covering spaces.
Theorem N21.1. is an injective homomorphism.
The subgroup determines the covering ( and ). Assuming is nice enough (“semi-locally simply connected”), for each subgroup of there is a covering .
Lifting lemmas for loops
Theorem N21.2. A loop in lifts to a loop in , i.e. for a loop at .
General lifting lemma
Theorem N21.3. Let be a covering map, path-connected and locally path-connected, and a continuous map with . Then can be lifted to a continuous with and . If the lift exists, it is unique.
(Margin note: I should create my own shorthand script specifically for math.)
Definition N21.4. is locally path-connected if for all and every neighborhood , there is a neighborhood of such that is path-connected.
Definition N21.5. Let and . Say and are equivalent if there is a homeomorphism such that (equivalently ).
Assume are path-connected and is locally path-connected (then so are ).