By J. P. May

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**Example text**

38 GRAPHS Proof. We may write G = F/N for some free group F and normal subgroup N . As above, we may realize the inclusion of N in F by passage to fundamental groups from a cover p : E −→ B. Define the (unreduced) cone on E to be CE = (E × I)/(E × {1}) and define X = B ∪p CE/(∼), where (e, 0) ∼ p(e). Let U and V be the images in X of B ∐ (E × [0, 3/4)) and E × (1/4, 1], respectively, and choose a basepoint in E × {1/2}. Since U and U ∩ V are homotopy equivalent to B and E via evident deformations and V is contractible, a consequence of the van Kampen theorem gives the conclusion.

Choose x ∈ X, let b = f (x), and choose e ∈ Fb . There exists a map g : X −→ E such that g(x) = e and p ◦ g = f if and only if f∗ (π1 (X, x)) ⊂ p∗ (π1 (E, e)) in π1 (B, b). When this condition holds, there is a unique such map g. Proof. If g exists, its properties directly imply that im(f∗ ) ⊂ im(p∗ ). Thus assume that im(f∗ ) ⊂ im(p∗ ). Applied to the covering Π(p) : Π(E) −→ Π(B), the analogue for groupoids gives a functor Π(X) −→ Π(E) that restricts on objects to the unique map g : X −→ E of sets such that g(x) = e and p ◦ g = f .

Define E to be the set of equivalence classes of paths f in B that start at b and define p : E −→ B by p[f ] = f (1). Of course, the equivalence relation is homotopy through paths from b to a given endpoint, so that p is well defined. Thus, as a set, E is just StΠ(B) (b), exactly as in the construction of the universal cover of Π(B). The topology of B has a basis consisting of path connected open subsets U such that π1 (U, u) −→ π1 (B, u) is trivial for all u ∈ U . Since every loop in U is equivalent in B to the trivial loop, any two paths u −→ u′ in such a U are equivalent in B.