Seifert–Van Kampen theorem

Unfortunately, the theorem as given above does not compute the fundamental group of the circle – which is the most important basic example in algebraic topology – because the circle cannot be realised as the union of two open sets with connected intersection. This problem can be resolved by working with the fundamental groupoid ${\displaystyle \pi _{1}(X,A)}$ on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points.[2]

This groupoid consists of homotopy classes relative to the end points of paths in X joining points of A ∩ X. In particular, if X is a contractible space, and A consists of two distinct points of X, then ${\displaystyle \pi _{1}(X,A)}$ is easily seen to be isomorphic to the groupoid often written ${\displaystyle {\mathcal {I}}}$ with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups.[3] The groupoid ${\displaystyle {\mathcal {I}}}$ also allows for groupoids a notion of homotopy: it is a unit interval object in the category of groupoids.

A connected union of two non connected spaces, with set of base points

The category of groupoids admits all colimits, and in particular all pushouts.

Theorem. Let the topological space X be covered by the interiors of two subspaces X₁, X₂ and let A be a set which meets each path component of X₁, X₂ and X₀ = X₁ ∩ X₂. Then A meets each path component of X and the diagram P of morphisms induced by inclusion

is a pushout diagram in the category of groupoids.[4]

This theorem gives the transition from topology to algebra, in determining completely the fundamental groupoid ${\displaystyle \pi _{1}(X,A)}$; one then has to use algebra and combinatorics to determine a fundamental group at some basepoint.

One interpretation of the theorem is that it computes homotopy 1-types. To see its utility, one can easily find cases where X is connected but is the union of the interiors of two subspaces, each with say 402 path components and whose intersection has say 1004 path components. The interpretation of this theorem as a calculational tool for "fundamental groups" needs some development of 'combinatorial groupoid theory'.[5][6] This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid ${\displaystyle {\mathcal {I}}}$ by identifying, in the category of groupoids, its two vertices.

There is a version of the last theorem when X is covered by the union of the interiors of a family ${\displaystyle \{U_{\lambda }:\lambda \in \Lambda \}}$ of subsets.[7][8]

The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets ${\displaystyle U_{\lambda }}$, then A meets all path components of X and the diagram

${\displaystyle \bigsqcup _{(\lambda ,\mu )\in \Lambda ^{2}}\pi _{1}(U_{\lambda }\cap U_{\mu },A)\rightrightarrows \bigsqcup _{\lambda \in \Lambda }\pi _{1}(U_{\lambda },A)\rightarrow \pi _{1}(X,A)}$

of morphisms induced by inclusions is a coequaliser in the category of groupoids.

[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points [...]

— Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation)

One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces. For example, consider the sphere ${\displaystyle S^{2}}$. Pick open sets ${\displaystyle A=S^{2}\setminus \{n\}}$ and ${\displaystyle B=S^{2}\setminus \{s\}}$ where n and s denote the north and south poles respectively. Then we have the property that A, B and A ∩ B are open path connected sets. Thus we can see that there is a commutative diagram including A ∩ B into A and B and then another inclusion from A and B into ${\displaystyle S^{2}}$ and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result

${\displaystyle \pi _{1}(S^{2})=\pi _{1}(A)\cdot \pi _{1}(B)/\ker(\Phi ).}$

However, A and B are both homeomorphic to R² which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of ${\displaystyle S^{2}}$ is trivial.

Given two pointed spaces ${\displaystyle (X,x)}$ and ${\displaystyle (Y,y)}$ we can form their wedge sum, ${\displaystyle (X\vee Y,p)}$, by taking the quotient of ${\displaystyle X\coprod Y}$ by identifying their two basepoints.

If ${\displaystyle x}$ admits a contractible open neighborhood ${\displaystyle U\subset X}$ and ${\displaystyle y}$ admits a contractible open neighborhood ${\displaystyle V\subset Y}$ (which is the case if, for instance, ${\displaystyle X}$ and ${\displaystyle Y}$ are CW complexes), then we can apply the Van Kampen theorem to ${\displaystyle X\vee Y}$ by taking ${\displaystyle X\vee V}$ and ${\displaystyle U\vee Y}$ as the two open sets and we conclude that the fundamental group of the wedge is the free product of the fundamental groups of the two spaces we started with:

${\displaystyle \pi _{1}(X\vee Y,p)\cong \pi _{1}(X,x)*\pi _{1}(Y,y)}$.

A more complicated example is the calculation of the fundamental group of a genus-n orientable surface S, otherwise known as the genus-n surface group. One can construct S using its standard fundamental polygon. For the first open set A, pick a disk within the center of the polygon. Pick B to be the complement in S of the center point of A. Then the intersection of A and B is an annulus, which is known to be homotopy equivalent to (and so has the same fundamental group as) a circle. Then ${\displaystyle \pi _{1}(A\cap B)=\pi _{1}(S^{1})}$, which is the integers, and ${\displaystyle \pi _{1}(A)=\pi _{1}(D^{2})={1}}$. Thus the inclusion of ${\displaystyle \pi _{1}(A\cap B)}$ into ${\displaystyle \pi _{1}(A)}$ sends any generator to the trivial element. However, the inclusion of ${\displaystyle \pi _{1}(A\cap B)}$ into ${\displaystyle \pi _{1}(B)}$ is not trivial. In order to understand this, first one must calculate ${\displaystyle \pi _{1}(B)}$. This is easily done as one can deformation retract B (which is S with one point deleted) onto the edges labeled by

${\displaystyle A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}.}$

This space is known to be the wedge sum of 2n circles (also called a bouquet of circles), which further is known to have fundamental group isomorphic to the free group with 2n generators, which in this case can be represented by the edges themselves: ${\displaystyle \{A_{1},B_{1},\dots ,A_{n},B_{n}\}}$. We now have enough information to apply Van Kampen's theorem. The generators are the loops ${\displaystyle \{A_{1},B_{1},\dots ,A_{n},B_{n}\}}$ (A is simply connected, so it contributes no generators) and there is exactly one relation:

${\displaystyle A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}=1.}$

Using generators and relations, this group is denoted

${\displaystyle \left\langle A_{1},B_{1},\dots ,A_{n},B_{n}\left|A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}\right.\right\rangle .}$

If X is space that can be written as the union of two open simply connected sets U and V with U ∩ V non-empty and path-connected, then X is simply connected.[10]