**Last time**:

**Lemma 24**: Fix basepoints as usual. There is a lift of such that if and only if

.

Furthermore, if the lift exists, it is unique.

**Lemma 25**: Given a choice of , there exists an elevation of at . Furthermore, is unique in the sense that if is another elevation of and then there is a homeomorphism and the diagram

commutes.

**Proof**:

Let be defined by . By Lemma 24, the composition

lifts at , call this lift . Suppose

Then . But by Lemma 24, . This implies .

Another, more categorical construction uses the **fibre product**.

The fibre product is defined by

.

There are obvious maps

given by forgetting factors.

**Exercise**: If is a covering map then is a covering map.

**Lemma 26**: Fix . Let and let for . Let , and let be the connected component containing . Then is an elevation of at , and every elevation of arises in this way.

**Proof**: To prove that is an elevation we just observe that . Now suppose

is an elevation. Then , with .

The covering map factors trhough , and so is a covering map. Because is an elevation, is a homeomorphism onto its image, a connected component of .

What has this got to do with graphs of spaces/groups?

Let be a vector space, and let be an edge map. Define to be the mapping cylinder

comes with a map such that and . This is an inclusion , and . Let be a covering map.

Let be the fibre product. There’s a map ; . Clearly, .

Therefore, is an injection. It’s easy to see that induces a bijection at the level .

**Lemma 27**: Any covering space arises as the fibre product of a covering map .

**Proof**: Let be a covering map

let be the fibre product of and .

Define by . As before, is a covering map.

## 2 comments

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30 November 2011 at 10.24 am

Ronnie BrownYou might like to look at the exposition of covering spaces and in particular fibre products, or pullbacks, in Chapter 10 of

R. Brown, _Topology and groupoids_, Booksurge, 2006 (available from amazon)

a revised version of editions with different titles published in 1968, 1988. This uses the notion of _covering morphism of groupoids_ and proves an equivalence of categories by the fundamental groupoid functor $\pi_1$

(covering spaces of $X$) $ \to $ (covering groupoids of $\pi_1 X$)

for spaces satisfying the usual local conditions. This also leads to a family of Mayer-Vietoris type exact sequences for a pullback of coverings, which contains the most common uses, including double cosets.

12 December 2011 at 5.13 am

Henry WiltonThanks for pointing this out, Ronnie. I think there are several different treatments of this material – there’s also a related paper by Bass. This arose out of reading the work of Wise.