§ Covering spaces
§ Covering spaces: Intuition
- Consider the map . This is a 2-to-1 map. We can try to define an inverse regardless.
- We do define a "square root" if we want. Cut out a half-line called for branch cuts. We get two functions on , such that . Here, we have .
- The point of taking the branch cut is to preserve simply connectedness. is not simply connected, while is simply connected! (This seems so crucial, why has no one told me this before?!)
- Eg 2: exponential. Pick . This is surjective, and infinite to 1. .
- Again, on , we have , such that .
- A covering map is, roughly speaking, something like the above. It's a map that's n-to-1, which has n local inverse defined on simply connected subsets of the target.
- So if we have , we have (for ) such that .
§ Covering spaces: Definition
- A subset is a called as an elementary neighbourhood if there is a discrete set and a homeomorphism such that or .
- Alternative definition A subset is called as evenly covered/elementary nbhd if where the are disjoint and open, and is a homeomorphism for all .
- An elementary neighbourhood is the region where we have the local inverses (the complement of a branch cut).
- We get for each , a map and then along sending .
- We say is a covering map if is covered by elementary neighbourhoods.
- We say is an elementary sheet if it is path connected and is an elementary neighbourhood.
- So, consider . If we cut the space at , then we will have elementary neighbourhood and elementary sheets .
- The point is that the inverse projection takes to some object of the form : a local product! So even though the global covering space does not look like a product of circles, it locally does. So it's some sort of fiber bundle?
Slogan: Covering space is locally disjoint copies of the original space.
§ Path lifting and Monodromy
- Monodromy is behaviour that's induced in the covering space, on moving in a loop in a base.
- Etymology: Mono --- single, drome --- running. So running in a single loop / running around a single time.
- Holonomy is a type of monodromy that occurs due to parallel transport in a loop, to detect curvature
- Loop on the base is an element of .
- Pick some point . Consider ( for fiber).
- Now move in a small loop on the base, . The local movement will cause movement of the elements of the fiber.
- Since , the elements of the fiber at the end of the movement are equal to the original set .
- So moving in a loop induces a permutation of the elements of the fiber .
- Every element of induces a permutation of elements of the fiber .
- This lets us detect non-triviality of . The action of on the fiber lets us "detect" what is.
- We will define what is means to "move the fiber along the path".
§ Path lifting lemma
Theorem :Suppose is a covering map. Let
be a path such that , and let [ is in the fiber of ].
Then there is a unique path which "lifts" .
That is, , such that .
Slogan: Paths can be lifted. Given how to begin the lift, can be extended all the way.
- Let be a collection of elementary neighbourhoods of .
- is an open cover (in the compactness sense) of .
- By compactness, find a finite subcover. Divide interval into subintervals such that lands in , an elementary neighbourhood.
- Build by induction on .
- We know that should be .
- Since we have an elementary neighbourhood, it means that there are a elementary sheets living over , indexed by some discrete set . lives in one of thse sheets. We have local inverses . One of them lands on the sheet of , call it . So we get a map such that .
- Define .
- Extend upto .
- Continue all the way upto .
- To get from , there exists a such that . Define .
- This is continuous because continuous by definition, continuous by neighbourhood, is pieced together such that endpoints fit, and is thus continuous.
- Can check this is a lift! We get . Since is a local inverse of , we get in the region.
§ 7.03: Path lifting: uniqueness
If we have a space and a covering space , for a path that
starts at , we can find a path which starts at
and projects down to : . We want to show
that this path lift is unique
Let be a covering space. Let be a connected space
Let be a continuous map (for us, ).
Let be lifts of ( , ).
We will show that iff the lifts are equal for some T.
Slogan: Lifts of paths are unique: if they agree at one point, they agree at all points!
- We just need to show that if and agree somewhere in , they agree everywhere. It is clear that if they agree everywhere, they must agree somewhere.
- To show this, pick the set where agree in : .
- We will show that is open and closed. Since is connected, must be either the full space or the empty set. Since is assumed to be non-empty, and the two functions agree everywhere.
- (Intuition: if both and are open, then we can build a function that colors in two colors continuously; ie, we can partition it continuously; ie the spaces must be disconnected. Since is connected, we cannot allow that to happen, hence or .)
- Let . Let be an evenly covered neighbourhood/elementary neighbourhood of downstairs (in ). Then we have such that is a local homeomorphism.
- Since are continuous, we will have opens in , which contain upstairs (mirrroring containing downstairs).
- The pre-images of , along give us open sets .
- Define . If , then and thus on all of . So, is open.
- If , then and thus on (since , and is injective within , ie within ). So is open.
- Hence we are done, as is non-empty and clopen and is thus equal to . Thus, the two functions agree on all of .
§ Homotopy lifting, Monodromy
- Given a loop in based at , the monodromy around is a permutation , where where is the unique lift of staring at . We have that .
- Claim: if then .
- We need a tool: homotopy lifting lemma.
Slogan: permutation of monodromy depends only on homotopy type
§ Homotopy lifting lemma/property of covering spaces
Suppose is a covering map and is a homotopy
of paths rel. endpoints ( and are independent of /
endpoints are fixed throughout the homotopy). Then there exists for each
(ie, ), a completion
of the lifted homotopy (ie, ).
Moreover, this lifted homotopy is rel endpoints: ie, the endpoints of are
independent of .
Slogan: homotopy lifted at 0 can be lifted for all time
- Let be the homotopy in such that . Subdivide the square into rectangles such that is contained in for some elementary neighbourhood . We build by building local inverses such that . We then set .