Covering

Main theorem of coverings

Let be a locally path-connected, connected, and semilocally simply connected topological space. Then for every subgroup there exists a covering unique up to equivalence with characteristic subgroup . homotopy

Construction

Take the universal covering and consider where is an isomorphism. The covering is given by with

Proof

Uniqueness up to equivalence follows from equivalence of coverings criterion. Since is semilocally simply connected, it has a universal covering . Let According to Deck transformation group of a regular covering as quotient

is an isomorphism, where is the unique lift of with , and denotes a unique deck transformation with this property.

Now take the orbit space with the canonical projection

Since the deck transformation group acts properly discontinuously, so too does , and the orbit space of a properly discontinuous effective group action forms a covering, which in this case is universal. Thus

We now define

which is well-defined since iff for some , and then ; and continuous by Universal property.

Now let and let be a neighbourhood of evenly covered by with sheets . Let such that for all there exists exactly one such that , and let . Then

and , therefore is a covering. Then by construction

so as required.

---

tidy | en | sembr