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If {M} is a connected topological manifold, and {p} is a point in {M}, the (topological) fundamental group {\pi_1(M,p)} of {M} at {p} is traditionally defined as the space of equivalence classes of loops starting and ending at {p}, 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 {\phi: N \rightarrow M}, the fundamental group {\pi_1(M,p)} acts (on the right) by monodromy on the fibre {\phi^{-1}(\{p\})}, and conversely given any discrete set with a right action of {\pi_1(M,p)}, 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 {M}, and the category of discrete {\pi_1(M,p)}-sets.

One of the basic tools used to compute fundamental groups is van Kampen’s theorem:

Theorem 1 (van Kampen’s theorem) Let {M_1, M_2} be connected open sets covering a connected topological manifold {M} with {M_1 \cap M_2} also connected, and let {p} be an element of {M_1 \cap M_2}. Then {\pi_1(M_1 \cup M_2,p)} is isomorphic to the amalgamated free product {\pi_1(M_1,p) *_{\pi_1(M_1\cap M_2,p)} \pi_1(M_2,p)}.

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 {M_1 \cup M_2}, carefully deforming them into combinations of loops in {M_1} or in {M_2} 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 {M_1} and a cover of {M_2} together to give a cover of {M}, 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.

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