If is a connected topological manifold, and is a point in , the (topological) fundamental group of at is traditionally defined as the space of equivalence classes of loops starting and ending at , with two loops considered equivalent if they are homotopic to each other. (One can of course define the fundamental group for more general classes of topological spaces, such as locally path connected spaces, but we will stick with topological manifolds in order to avoid pathologies.) As the name suggests, it is one of the most basic topological invariants of a manifold, which among other things can be used to classify the covering spaces of that manifold. Indeed, given any such covering , the fundamental group acts (on the right) by monodromy on the fibre , and conversely given any discrete set with a right action of , one can find a covering space with that monodromy action (this can be done by “tensoring” the universal cover with the given action, as illustrated below the fold). In more category-theoretic terms: monodromy produces an equivalence of categories between the category of covers of , and the category of discrete -sets.
One of the basic tools used to compute fundamental groups is van Kampen’s theorem:
Theorem 1 (van Kampen’s theorem) Let be connected open sets covering a connected topological manifold with also connected, and let be an element of . Then is isomorphic to the amalgamated free product .
Since the topological fundamental group is customarily defined using loops, it is not surprising that many proofs of van Kampen’s theorem (e.g. the one in Hatcher’s text) proceed by an analysis of the loops in , carefully deforming them into combinations of loops in or in and using the combinatorial description of the amalgamated free product (which was discussed in this previous blog post). But I recently learned (thanks to the responses to this recent MathOverflow question of mine) that by using the above-mentioned equivalence of categories, one can convert statements about fundamental groups to statements about coverings. In particular, van Kampen’s theorem turns out to be equivalent to a basic statement about how to glue a cover of and a cover of together to give a cover of , and the amalgamated free product emerges through its categorical definition as a coproduct, rather than through its combinatorial description. One advantage of this alternate proof is that it can be extended to other contexts (such as the étale fundamental groups of varieties or schemes) in which the concept of a path or loop is no longer useful, but for which the notion of a covering is still important. I am thus recording (mostly for my own benefit) the covering-based proof of van Kampen’s theorem in the topological setting below the fold.
— 1. Proof of van Kampen theorem —
The proof of van Kampen’s theorem boils down (after using the above-mentioned equivalence of categories between covers of a manifold , and sets with an action of the fundamental group) to the following fact about covers:
Proposition 2 (Gluing of covers) Let be connected open sets covering a connected topological manifold with also connected, and let be an element of . If and are covering maps which become isomorphic upon restricting the base to , then there is a covering map which becomes isomorphic to on restricting the base to , and isomorphic to on restricting the base to (and with all four isomorphisms forming a commuting square).
This proposition is easily verified by gluing together and as topological spaces along the indicated isomorphism between and , and checking that the resulting space is still a covering space.
Now we can prove van Kampen’s theorem. Suppose that one has group homomorphisms , to a target group which form a commuting square with the canonical homomorphisms from to and . It will suffice to show that there is a unique homomorphism such that factors as the composition of with the canonical homomorphism from to for .
We first demonstrate existence. Let be a universal cover for , thus is simply connected, and if we pick a base point , then every other point in that fibre there is a unique element for which , where is the (right) action of by monodromy on . This gives a left action of on by deck transformations for each , which maps to for any :
The fibres of the universal cover are copies of . We can now form a new cover whose fibres are copies of , by first forming the Cartesian product (which still covers ) and then quotienting out by the equivalence for all , , . This is a (possibly disconnected) covering space for , whose fibre above can be identified with by identifying with the equivalence class . The monodromy (right) action of on this fibre is then identified with the right action of on induced by . Furthermore, the group acts on on the left by deck transformations, with each mapping to .
Similarly, we can form a covering of whose monodromy action of on the fibre can be identified with the right action of on induced by . When both of these covering spaces are restricted to , the monodromy action of on the fibre above are then isomorphic to each other, and thus the two restrictions are isomorphic to each other too. By Proposition 2, we can then glue these two covers together to obtain a cover which is isomorphic to or upon restricting the base to or respectively, with all four isomorphisms forming a commuting square; in particular, the fibre is still identified with . The monodromy right action of on then restricts to the previously described action of on and of on . Also, because acted on the left by deck transformations on both and , in a manner which can be seen to be compatible with the isomorphism on restriction to , continues to act by deck transformations on the gluid cover . As monodromy actions commute with deck transformations, we conclude that the right action of an element of on is given by right multiplication by an element of . It is then routine to verify that is a homomorphism with the required properties.
Now we prove uniqueness. It suffices to show that is generated as a group by the images of and . Suppose this were not the case, so that the images of and generate a proper subgroup of . Let be a universal cover of , so that the fibre may be identified with (after fixing a reference point in that fibre as before). Let be the restriction of to . The monodromy (right) action of on the fibre above is induced by the canonical homomorphism from to . In particular, is preserved by this action, and so one can find a proper subcover of whose fibre corresponds to . Similarly, we can find a proper subcover of the restriction of to with the same fibre. By Proposition 2, we may glue these two covers together to obtain a proper subcover of . But such a proper subcover cannot exist because is connected, and the claim follows.
Remark 1 The arguments above relied heavily on the universal cover and its attendant deck transformations and monodromy actions, mostly in order to give a concrete equivalence between coverings of and discrete sets with actions of the fundamental group . It is also possible to proceed without constructing this cover, working instead with a directed family of Galois covers as a substitute for the universal cover to obtain this equivalence of categories; this is the approach in Grothendieck’s Galois theory, which among other things can be used to construct the étale fundamental group, as this is a context in which universal covers need not exist, but one still has plenty of Galois covers. See for instance Szamuely’s text for more details.