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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.
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30 November 2011 at 10.24 am
Ronnie Brown
You 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 Wilton
Thanks 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.