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Math · Advanced Math
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2. Prove the following: Theorem. IfG is a finitely presented group, then there is a compact Hausdorff space X whose fundamental group is isomorphic to G Proof. Suppose G has a presentation consisting of n generators and m relations. Let A be the wedge of n circles; form an adjunction space X from the union of A and m copies Bi, Bm of the unit ball by means of a continuous map (a) Show that X is Hausdorff. (b) Prove the theorem in the case m-1. (c) Proceed by induction on m, using the algebraic result stated in the following exercise. The construction outlined in this exercise is a standard one in algebraic topo ogy; the space X is called a two-dimensional CW complex.

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