Covering

Lift of a map to a covering space

Let and be a topological spaces, be a covering of with , and be a continuous map. A lift of is any function so that , #m/def/topology i.e. the following diagram commutes in :

https://q.uiver.app/#q=WzAsMyxbMiwyLCJYIl0sWzIsMCwiXFx0aWxkZSBYIl0sWzAsMiwiWSJdLFsyLDAsImYiXSxbMSwwLCJwIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMSwiXFx0aWxkZSBmIl1d

Lifts fill a fundamental role in Homotopy theory MOC, in particular they allow for the computation of the Fundamental group. Their usefulness follows from the main theorem below.

Main theorem

Let be a connected covering, be a connected and locally path-connected1 space, and be a morphism in . Then there exists a lift of iff #m/thm/topology

i.e. the image of is a subset of the image2 of , where is the Fundamental group functor. Furthermore if exists it is unique.

Construction of lift

A lift of is constructed as follows: For each , by path-connectedness there exists a path from to . Then define a path in , which by Second lemma Lifts of paths has a unique lift with . Then let .

The proof involves four lemmas, each relying on the previous: Uniqueness may be proven immediately, then we prove the special cases of lifts of paths and lifts of homotopies of paths, and then the requirement given for the fundamental group.

First lemma: Uniqueness

Let be a connected covering, be a connected space, be a continuous function, and be lifts of . Then iff for some . #m/thm/topology

Proof

The reverse direction is obvious. For the forward direction, consider the set

Let , , and be an evenly covered open neighbourhood of . Let be the sheets over containing and respectively. Then is an open neighbourhood of , and

Now if , then and consequently

Thus is the union of all obtained from with and is thus open. Now if , then and consequently for all . Thus is the union of all obtained from with and is thus open. Therefore is clopen and inhabited ( ), so since is connected, . Hence .

Second lemma: Lifts of paths

Let be a connected covering and be a continuous path from . For each there exists exactly one lifted path from . #m/thm/topology

Proof

Uniqueness follows from First lemma Uniqueness, but is also self-evident in the following argument. For each , let be an evenly covered open neighbourhood of . Then is an open cover of , and is an open cover of . Using a Lebesgue number may be evenly subdivided with

so that where for all . Now consider a lift . Clearly , where is the sheet over containing . Thus if is set, is unique and well-defined.

Third lemma: Lifts of homotopies of paths

Let be a connected covering and be continuous paths with the same endpoints homotopic to one another via . Let and be the unique lifts of respectively with . Then there exists a unique lift of the homotopy , and in particular . #m/thm/homotopy

Proof

First, notice that if a lift of with exists, it is necessarily unique (by First lemma Uniqueness) and a homotopy from to : Clearly and for all , and since and are discrete, both and must be constant for all , so and we let . By construction is a homotopy from to , but is a lift of with for each , thus by uniqueness in particular for , and hence is the desired homotopy.

For existence we use a similar construction to above: Using a Lebesgue number argument may be subdivided into a grid with

so that for each square with , its image is contained entirely within an evenly covered open set in , i.e. . Now consider a lift . If the bottom left corner is set, then clearly , where is the sheet over containing . Then by Second lemma Lifts of paths the edges automatically agree, thus by starting with we obtain a well-defined, unique lift of .

Fourth lemma: Condition for the existence of a lift

Let be a connected covering, be a path-connected space, and be a morphism in . If a lift exists, then . #m/thm/homotopy

Proof

Since , it follows from functor properties of the Fundamental group that , and thus the image of must be contained within the image of .

Proof of main theorem

The forward direction follows from Fourth lemma Condition for the existence of a lift.

Proof the construction is well-definined

It remains to show that is independent from the choice of path . To this end let be a path from to . Then is a continuous loop with basepoint . Since is guaranteed a lift by Second lemma Lifts of paths, it follows , and thus there exists a continuous loop with basepoint such that . Let , , and , so and thus . Then if and are the lifts of and respectively with , then is the lift of . Hence by Third lemma Lifts of homotopies of paths, , and in particular . Hence is the same regardless of the selected path .

#state/tidy | #lang/en | #SemBr

Footnotes

  1. And thus path-connected, since A locally path-connected space is path-connected iff it is connected ↩

  2. In Algebraische Topologie wird dies als die charakteristische Untergruppe bezeichnet (p. 91). ↩