By K. Ueno

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Fix a basepoint b ∈ B. We turn the properties of paths that must hold in a universal cover into a construction. 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 .

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 . We need only check that g is continuous, and this holds because p is a local homeomorphism. In detail, if y ∈ X and g(y) ∈ U , where U is an open subset of E, then there is a smaller open neighborhood U of g(y) that p maps homeomorphically onto an open subset V of B.

If G has k generators, then χ(B) = 1 − k. If [G : H] = n, then Fb has cardinality n and χ(E) = nχ(B). Therefore 1 − χ(E) = 1 − n + nk. We can extend the idea to realize any group as the fundamental group of some connected space. Theorem. For any group G, there is a connected space X such that π1 (X) is isomorphic to G. 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.

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An Introduction to Algebraic Geometry by K. Ueno

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