the Seifert-van Kampen theorem
/ ZY-fert van KAM-pen /
If you know the fundamental group of two overlapping pieces of a space, can you compute the fundamental group of the whole? Yes — the Seifert-van Kampen theorem is the tool that glues local pi_1 information into global pi_1. It is the homotopy-theoretic analogue of computing the behavior of a machine from the behavior of its overlapping subassemblies, and it is how almost every fundamental group of a built-up space actually gets calculated.
Suppose X = U union V with U, V, and U ∩ V all open and path-connected, and pick a basepoint x_0 in U ∩ V. The theorem says pi_1(X, x_0) is the free product of pi_1(U) and pi_1(V) amalgamated over pi_1(U ∩ V): in symbols, pi_1(X) = pi_1(U) *_{pi_1(U ∩ V)} pi_1(V). Concretely, take generators and relations (a presentation) for pi_1(U) and for pi_1(V), throw them together, and add one new relation for each generator of pi_1(U ∩ V) saying 'its image coming in via U equals its image coming in via V.' The intuition: a loop in X can be chopped into arcs each living in U or in V (by compactness and the Lebesgue number), so loops are words in the two groups, and the only relations beyond those of U and V come from loops that lie in the overlap and can be read either way.
This single theorem computes a huge swath of examples. The wedge of two circles has pi_1 = Z * Z (free on two generators), since the overlap is contractible and contributes no relations. A closed orientable surface of genus g has the famous presentation pi_1 = < a_1, b_1, ..., a_g, b_g | product of commutators [a_i, b_i] = 1 >, read off by cutting the surface into a disk glued along its boundary word. Two honest cautions: the open-cover and path-connected-overlap hypotheses matter — a disconnected intersection forces the groupoid version of the theorem, not the simple amalgamated-product form — and the theorem computes pi_1 only, telling you nothing directly about higher pi_n, which do NOT obey any such van Kampen gluing.
Compute pi_1 of the Klein bottle. Present it as a square with edge word a b a b^{-1}; take U a neighborhood of the 1-skeleton (a wedge of two circles, pi_1 free on a, b) and V the open 2-cell (contractible). The overlap is an annulus whose loop maps to the attaching word, so pi_1 = < a, b | a b a b^{-1} = 1 > = < a, b | a b a = b >.
Van Kampen turns a polygon's edge-gluing word into a group presentation — here the Klein bottle's pi_1.
The clean amalgamated-product statement needs U ∩ V path-connected; if the overlap has several components you must use the fundamental groupoid version, or the formula simply does not apply. Also, the theorem gives a presentation, and deciding whether two presentations describe isomorphic groups can itself be undecidable.