Pushout (category Theory) - Application: The Seifert-van Kampen Theorem

Application: The Seifert-van Kampen Theorem

Returning to topology, the Seifert-van Kampen theorem answers the following question. Suppose we have a path-connected space X, covered by path-connected open subspaces A and B whose intersection is also path-connected. (Assume also that the basepoint * lies in the intersection of A and B.) If we know the fundamental groups of A, B, and their intersection D, can we recover the fundamental group of X? The answer is yes, provided we also know the induced homomorphisms and The theorem then says that the fundamental group of X is the pushout of these two induced maps. Of course, X is the pushout of the two inclusion maps of D into A and B. Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when D is simply connected, since then both homomorphisms above have trivial domain. Indeed this is the case, since then the pushout (of groups) reduces to the free product, which is the coproduct in the category of groups. In a most general case we will be speaking of a free product with amalgamation.

There is a detailed exposition of this, in a slightly more general setting (covering groupoids) in the book by J. P. May listed in the references.