yet the simplest known proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore, its fundamental group H is free. On the other hand, this type of application is also handled more simply by the use of covering morphisms of groupoids, and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology; see . Topological combinatorics. Notable algebraic topologists Important theorems in algebraic topology See also Notes References (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets). . (Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids). . "Gives a general theorem on the fundamental groupoid with a set of base points of a space which is the union of open sets." . . "The first 2-dimensional version of van Kampen's theorem." This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in singular homology, or the method of simplicial approximation. It contains a lot of material on crossed modules. . A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper. . A modern, geometrically flavoured introduction to algebraic topology. . Further reading and . Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.