Contents

Idea

The Seifert-van Kampen theorem is a classical theorem in algebraic topology which computes the fundamental group of a topological space in terms of a decomposition into open subsets.

It is most naturally expressed by saying that the fundamental groupoid functor preserves certain colimits. Here there is a bifurcation in possible generalizations, however. The colimits of spaces we consider (covers by open subsets) are both strict colimits and homotopy colimits. But we can ask the resulting colimit of groupoids to be either a homotopy colimit (i.e. a 2-colimit) or a strict colimit. The first is more natural for formal work; the second is more useful for computation. Accordingly, we will speak of homotopy van Kampen theorems and strict van Kampen theorems.

Statement

For fundamental groups

The most classical version of the Seifert-van Kampen theorem for the fundamental group is the following.

From groups to groupoids

We can regard the classical theorem as being a statement about fundamental groupoids as follows. Recall that any group $G$ can be considered as a 1-object groupoid $BG$ by delooping. It is easy to see that the functor $B$ takes colimits of groups to strict colimits of groupoids. Moreover, a strict pushout of one-object groupoids is also a 2-pushout, because the maps we are pushing out are bijective on objects, hence cofibrations in the canonical model structure on $\mathrm{Gpd}$. Thus, under the hypotheses of the classical van Kampen theorem, we have a square

which is both a strict pushout and a 2-pushout in $\mathrm{Gpd}$.

Now let $X\prime$ be the path component? of $X$ which contains $U\cap V$ (there is exactly one such, because $U\cap V$ is path-connected), and similarly $U\prime$ and $V\prime$. Then the fundamental groupoid ${\Pi }_1(X\prime )$ is equivalent to $B{\pi }_1(X,x)$, and similarly in the other cases. Since 2-pushouts are invariant under equivalence of groupoids, we also have a 2-pushout square

This is an instance of what we referred to above as a homotopy van Kampen theorem. But we can easily make it a strict van Kampen theorem as well.

Note that the functors ${\Pi }_1(U\cap V)\to {\Pi }_1(U\prime )$ and ${\Pi }_1(U\cap V)\to {\Pi }_1(U\prime )$ are injective on objects, hence cofibrations in the canonical model structure, and so their strict pushout is also a 2-pushout. This strict pushout is therefore equivalent to ${\Pi }_1(X\prime )$. (It need not be isomorphic to it, but this can be remedied by defining the fundamental groupoid on a set of specified basepoints as below.)

We now consider more general versions of the van Kampen theorem which do not require $U\cap V$ to be path-connected, and which compute all of ${\Pi }_1(X)$ rather than just ${\Pi }_1(X\prime )$.

Strict van Kampen theorem for groupoids

Let us define ${\Pi }_1(X,X_0)$ of a space $X$ and a subset $X_0\subseteq X$ to be the full subgroupoid of ${\Pi }_1(X)$ on the set $X_0$. More generally, if $U$ is given as a subspace of $X$ and $X_0\subseteq X$, we write ${\Pi }_1(U,X_0)$ for ${\Pi }_1(U,X_0\cap U)$.

If $X_{*}=(X,X_0)$ is a pair consisting of a space $X$ and a set $X_0$ of base points, we say $X_{*}$ is connected if $X_0$ meets each path component of $X$. This is clearly a necessary and sufficient condition for ${\Pi }_1(X,X_0)$ to be equivalent to ${\Pi }_1(X)$.

Now let $\{U^i\to X\}$ for $i\in I$ be an open cover of $X$. If $d=(i,j)\in I^2$ we write $U^d$ for $U^i\cap U^j$, and similarly for $n$-fold intersections. For any $n$-tuple $e\in I^n$, let $U_{*}^e$ be the pair $(U^e,X_0\cap U^e)$. We then have a coequaliser diagram of pairs of spaces, where $a,b,c$ are determined by inclusions:

The limit “3” on the connectivity hypothesis may be analyzed as ”$1+2$”, where $1$ is the categorical dimension of the fundamental groupoids we are considering and $2$ is a universal constant. This becomes evident when generalizing to higher homotopy van Kampen theorems.

In one proof of the theorem (due to Brown and Razak, see below), the number 3 arises from the Lebesgue covering dimension of ${ℝ}^2$. This proof verifies the universal property for a coequaliser, using basic techniques: subdivide a path; deform a subdivision so that it is product of paths joining points of $X_0$; subdivide a homotopy rel end points, and deform this subdivision so that all subpaths join points of $X_0$; any composition of commutative squares in a groupoid is commutative.

In another, more categorical proof explained in the next section, the number $3$ arises from the fact that 2-colimits can be computed using coproducts and descent objects of 2-truncated simplicial objects.

Note that the theorem about groupoids does not reduce to a theorem about groups. For example, $X$ may be a connected space which is the union of two open sets each with many components and whose intersection has many path components. Such examples arise commonly from applications to combinatorial group theory; consider for example a covering space of a wedge of spaces. Or the connected $X$ may be the union of 23 open sets whose three fold intersections have 123 path components. In each case the fundamental group one might want to calculate is in the middle of this complicated combinatorial situation, but at least the theorem has turned a topological problem into a group theory and combinatorial problem, and the remarkable fact is that the fundamental groups are completely determined by the theorem. This is an “anomaly” for traditional algebraic topology, where invariants relating adjacent dimensions may be determined by exact sequences which do not give complete information. The reason for the success seems to be that groupoids have structure in dimensions 0 and 1, and so can model the gluing of spaces.

This suggests that higher homotopy van Kampen theorems could give new kinds of homotopical information, i.e. colimit theorems for higher homotopy invariants, which has proved to be so.

Homotopy van Kampen theorem for groupoids

From a 2-categorical point of view, the strict van Kampen theorem above can be decomposed into the following parts:

1. When $X$ is the union of a family of open sets, then it is the homotopy colimit of a diagram involving these open sets.

2. The fundamental groupoid functor preserves homotopy colimits (i.e. takes them to homotopy colimits, or equivalently 2-colimits, of groupoids).

3. Certain 2-colimits of groupoids can be computed over “2-truncated” subdiagrams.

4. In certain situations, strict colimits of groupoids are also 2-colimits.

We can then prove the theorem by proving each of these three parts separately. This is essentially accomplished in Farjoun.

Examples

Let $X=S^1$, $U=S^1\setminus \{N\}$, and $U=S^1\setminus \{S\}$, where $N$ and $S$ are antipodal points. Then $U\cong V\cong (0,1)$ are contractible, while $U\cap V\cong (0,1)\bigsqcup (0,1)$. Thus the homotopy van Kampen theorem gives a 2-pushout of groupoids that is equivalent to

It’s not hard to compute that this 2-pushout must be $B\mathbb{Z}$. But we can also obtain this directly from the strict van Kampen theorem with a set $X_0=\{u,v\}$ of two basepoints with $u\in U$ and $v\in V$, which yields a strict pushout of groupoids

where $ℐ$ is the invertible interval category, which is (literally) ${\Pi }_1(U,X_0)$ and ${\Pi }_1(V,X_0)$.

Note that the version of the theorem for fundamental groups does not apply, since $U\cap V$ is not path-connected.

Generalizations

• van Kampen theorem for toposes

• higher homotopy van Kampen theorem

References

The following paper gives the general theorem for the fundamental groupoid on a set of base points:

• Ronnie Brown and A. Razak, A van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85-88.

This version for pushouts is discussed in some detail in

• Ronnie Brown, Philip Higgins, Rafael Sivera, Nonabelian Algebraic Topology

as a background to the following chapters on versions for crossed modules and crossed complexes, i.e. for higher dimensional versions.

The proof via homotopy colimits is roughly taken from

• Emmanuel Dror Farjoun, Fundamental group of homotopy colimits, Adv. in Math. 182 (2004), 1-27. http://www.maths.ed.ac.uk/~aar/papers/farjoun.pdf

Other references cited above:

• Dan Dugger and Daniel Isaksen, Hypercovers in topology (pdf)