The Van Kampen Theorem

We now discuss another way of calculating the fundamental group of a space. This corresponds to the second general family of identification spaces we discussed, namely spaces which are obtained by glueing together two spaces along a common subspace. Before we can state the theorem we need some group theoretic preliminaries again.

Given two groups G and H containing a common subgroup K, one may wonder whether it is possible to ``glue'' G and H together along K. To make this precise, assume that there are two monomorphisms f : K $\longrightarrow$ G and g : K $\longrightarrow$ H. We want to construst a new group P and maps p : G $\longrightarrow$ P and q : H $\longrightarrow$ P, such that whenever L is an arbitrary group and $\alpha$ : G $\longrightarrow$ L, $\beta$ : H $\longrightarrow$ L are homomorphisms such that $\alpha$f = $\beta$g, there exists a unique homomorphism $\gamma$ : P $\longrightarrow$ L such that p$\gamma$ = $\alpha$ and q$\gamma$ = $\beta$.

To begin with consider the simplest example of this situation, namely where K = {1} and the maps f and g are the obvious ones. In that case the requirements above are that whenever $\alpha$ and $\beta$ are arbitrary homomorphisms into L there is a unique map from the group P into L, which factors both $\alpha$ and $\beta$. The group P in that case is easy to construct. Namely, consider the group whose elements are arbitrary words in alphabet from G and H and their inverses. Multiplication is given by juxtaposition and the only cancelation rules come from G and H. This group is called the free product of G and H and is denoted G*H. Before proceeding we discuss some examples.

Example 7.10   Let G = H = Z/2Z and let $\sigma$ and $\tau$ denote the generators of G and H respectiely. Then the group G*H is the collection of all words of the following forms 1, $\sigma$, $\tau$ , $\sigma$*$\tau$, $\tau$*$\sigma$, $\sigma$*$\tau$*$\sigma$, $\tau$*$\sigma$*$\tau$, etc. Multiplication is given by juxtaposition, thus for instance ($\sigma$*$\tau$)*($\sigma$*$\tau$*$\sigma$) = $\sigma$*$\tau$*$\sigma$*$\tau$*$\sigma$ and ($\sigma$*$\tau$)*($\tau$*$\sigma$*$\tau$) = $\tau$.

The free product of G and H has the feature that if K is an arbitrary group and there are homomorphisms f : G $\longrightarrow$ K and g : H $\longrightarrow$ K then there is a unique homomorphism from G*H to K which agrees with f and g on the repective images of G and H in the free product. In some sense the free product of two groups is in the world of groups corresponds to the wedge operation in the world of pointed spaces and to disjoint union for unpointed spaces. We have sketched a proof in the tutorials of the fact that the fundamental group of a wedge of two pointed spaces is the free product of the respective fundamental groups. The Van Kampen theorem is just a generalisation of this obsetvation.

Definition 7.11   Let G, H and K be groups and let homomorphisms g : K $\longrightarrow$ G and h : K $\longrightarrow$ H be given. The free amalgamated product of G and H over K is defined to be the quotient group of the free product G*H by the minimal normal subgroup containing all elements of the form g(k)*h(k)^-1.

Example 7.12   The free product of G and H is the same as the free amalgamated product over the trivial group with g and h being the obvious homomorphisms.

Many times free amalgamated products are easy to write in terms of generators and relations if a presentation for the groups involved is known.

Example 7.13   Let G = Z/4Z and H = Z/(Z6), generated by x and y respectively. Let K = Z/2Z generated by t and let g(t) = x² and h(t) = y³ be the unique non-trivial homomorphisms from K into G and H repsetively. Then

G*_KH = Z/4Z*_Z/2ZZ/6Z = $$\langle$$x, y  |  x⁴ = y⁶ = 1, x² = y³$$\rangle$$.

Example 7.14   Let G, H and K be as above but replace the homomorphism h by the trivial homomorphism. Then

G*_KH = $$\langle$$x, y  |  x⁴ = y⁶ = 1, x² = 1$$\rangle$$ = $$\langle$$x, y  |  x² = y⁶ = 1$$\rangle$$ = Z/2Z*Z/6Z.

Definition 7.15   Let G be a group. The abelianisation G_ab of G is the maximal abelian quotient group of G. It is obtained as the quotient group of G by the minimal normal subgroup containing all elements of the form x^-1y^-1xy for x, y $\in$ G.

Notice that two isomorphic groups have isomorphic abelianisations. However, two groups with the same abelianisation may not be isomorphic. In particular the abelianisation of an abelian group G is G itself because every element of the form x^-1y^-1xy is trivial. Thus (G_ab)_ab = G_ab but G $\neq$ G_ab unless G is abelian.

If G is given in terms of generators and relations then G_ab is presented by the same generators and for realtions one takes the original relations together with extra relations x^-1y^-1xy = 1, one for every pair of generators x and y. The reader might like to use the idea of abelianisation to convince himself that the two groups obtained above as free amalgamated products of Z/4Z by Z/6Z over Z/2Z are not isomorphic. The abelianisation of the first is Z/12Z, whereas the second abelianises to Z/2Z x Z/6Z.

Example 7.16   Let F₁, F₂ be free groups on two generators x, y and u, v respectively. Let K = Z denote the group of integers (free group on one generator) generated by z. Let f_i : K $\longrightarrow$ F_i be defined by f₁(z) = xyx^-1y^-1 and f₂(z) = uvu^-1v. The free amalgamated product has a presentation

F₁*_KF₂ = $$\langle$$x, y, u, v  |  xyx^-1y^-1 = uvu^-1v$$\rangle$$.

The free amalgamated product of G and H over K has the following property. Let F be an arbitrary group and let $\alpha$ : G $\longrightarrow$ F and $\beta$ : H $\longrightarrow$ F be homomorphisms such that $\alpha$g = $\beta$h. Let i : G $\longrightarrow$ G*_KH and j : H $\longrightarrow$ G*_KH denote the inclusions. Then there exists a unique homomorphism $\gamma$ : G*_KH $\longrightarrow$ F such that $\gamma$i = $\alpha$ and $\gamma$j = $\beta$. We are now ready to state the Van Kampen Theorem.

Theorem 7.17   Let X = U $\cup$ V with U, V and U $\cap$ V non-empty, open in X and path-connected. Assume the base point x₀ is in U $\cap$ V. Let u : U $\longrightarrow$ X and v : V $\longrightarrow$ X denote the inclusions. Then the induced maps on fundamental groups induce an isomorphism

$$\pi_{1}^{}$$(X, x₀) $$\cong$$ $$\pi_{1}^{}$$(U, x_o)*_{$\pi_{1}$(U $\cap$ V, x0)}$$\pi_{1}^{}$$(V, x₀).

Before we prove the theorem we explain its statement and examine a few examples. The theorem states the following. Under the topological assumptions on U and V, consider the homomorphisms g : $\pi_{1}^{}$(U $\cap$ V, x₀) $\longrightarrow$ $\pi_{1}^{}$(U, x₀) and f : $\pi_{1}^{}$(U $\cap$ V, x₀) $\longrightarrow$ $\pi_{1}^{}$(V, x₀). With respect to those, one can construct the free amalgamated produc of $\pi_{1}^{}$(U, x₀) and $\pi_{1}^{}$(V, x₀) over $\pi_{1}^{}$(U $\cap$ V, x₀). Now, there are the maps u_# : $\pi_{1}^{}$(U, x₀) $\longrightarrow$ $\pi_{1}^{}$(X, x₀) and v_# : $\pi_{1}^{}$(V, x₀) $\longrightarrow$ $\pi_{1}^{}$(X, x₀) and by the mapping property of the free amalgamated product there exists a unique homomorphism from $\gamma$ : $\pi_{1}^{}$(U, x_o)*_{$\pi_{1}$(U $\cap$ V, x0)}$\pi_{1}^{}$(V, x₀) to $\pi_{1}^{}$(X, x₀) such that $\gamma$i = u_# and $\gamma$j = v_#. The theorem states that $\gamma$ is an isomorphism.

Let us now consider a few examples.

Example 7.18   Let X and Y be pointed spaces. Then the Van Kampen theorem gives an indirect proof that $\pi_{1}^{}$(X $\vee$ Y,(x₀, y₀)) $\cong$ $\pi_{1}^{}$(X, x₀)*$\pi_{1}^{}$(Y, y₀).

The next family of examples come from connected sums of two closed surfaces. Recall that the connected sum construction is formed by removing an open disk from the respective surfaces and glueing them along the boundaries. Thus the connected sum of two surfaces can be though of as a union of two punctured surfaces intersecting on a circle. By thickening the circle of intersection a bit one can guarantee that the conditions of the theorem are satisfied. The next step is to compute the homomorphism induced by inclusion of the circle as the boundary of the respective punctured surface.

The first and easiest is the case where the surface is a 2-sphere. In that case the punctured surface is contractible and the induced map

$$\pi_{1}^{}$$(S¹) $$\longrightarrow$$ $$\pi_{1}^{}$$(S² $$\setminus$$ D²)

is trivial.

Corollary 7.19   The 2-sphere is simply-connected.

Next consider the torus T². We saw how the torus can be obtained from a square by identifying each pair of parallel edges ini the same orientation.

Lemma 7.20   The punctured torus T² $\setminus$ D² has the homotopy type of a wedge of two circles and the inclusion of the boundary S¹ $\longrightarrow$ T² $\setminus$ D² induces on fundamental groups the map which takes a generator of $\pi_{1}^{}$(S¹) to the element a^-1b^-1ab where $\pi_{1}^{}$(T² $\setminus$ D²) is the free group on a and b.

Remark 7.21   Notice that one may choose each one of the vertices of the square as a starting point (as a base point they are all the same) and also orient the identifications four different ways. This will result only in modifying the set of generators for $\pi_{1}^{}$(T² $\setminus$ D²).

Next we present a calculation of the fundamental group of the torus using the Van Kampen theorem.

Corollary 7.22   $\pi_{1}^{}$(T²) $\cong$ Z x Z.

The Klein bottle K² can be obtained from a square by identifying one pair of parallel edges in the same orientation and the other in opposite orientation.

Lemma 7.23   The punctured Klein bottle K² $\setminus$ D² has the homotopy type of a wedge of two circles and the inclusion of the boundary S¹ $\longrightarrow$ K² $\setminus$ D² induces on fundamental groups the map which takes a generator of $\pi_{1}^{}$(S¹) to the element a^-1b^-1ab^-1 where $\pi_{1}^{}$(K² $\setminus$ D²) is the free group on a and b.

Similar to the calculation of $\pi_{1}^{}$(T²) we now present a calculation of $\pi_{1}^{}$(K²) using the Van Kampen theorem.

Corollary 7.24   $\pi_{1}^{}$(K²) $\cong$ $\langle$a, b  |  a^-1b^-1ab^-1 = 1$\rangle$.

Now at last we are able to show easily that the torus and the Klein bottle are not homeomorphic. In fact they are not even homotopy equivalent.

Corollary 7.25   The torus and the Klein bottle have distinct homotopy type.

The real projective plane RP² is the orbit space of the antipodal action on the 2-sphere.

Lemma 7.26   The punctured real projective plane RP² $\setminus$ D² has the homotopy type of a circle and the inclusion of the boundary S¹ $\longrightarrow$ RP² $\setminus$ D² induces on fundamental groups the map which takes a generator of $\pi_{1}^{}$(S¹) to the element a² where $\pi_{1}^{}$(RP² $\setminus$ D²) generated by a.

Corollary 7.27   $\pi_{1}^{}$(RP²) $\cong$ Z/2Z

Corollary 7.28   The homotopy type of RP² is distinct from those of T² and K².

We are now ready to glue surfaces together. This will be done in a rather random way. Namely, we will not try to be systematic in the way we form connected sums of surfaces and will use various combinations of the 2-sphere, torus, Klein bottle and the real projective plane. Later on we will consider the classification theorem for closed surfaces and will in particular be able to present a calculation using the Van Kampen theorem of all possible fundamental groups of closed surfaces. This will enable us to argue that all the surfaces described in the clasification theorem have distinct homotopy types.

Lemma 7.29  

$$\pi_{1}^{}$$(T²#T²) $$\cong$$ $$\langle$$a, b, c, d  |  a^-1b^-1ab = c^-1d^-1cd$$\rangle$$.

Lemma 7.30  

$$\pi_{1}^{}$$(T²#K²) $$\cong$$ $$\langle$$a, b, c, d  |  a^-1b^-1ab = c^-1d^-1cd^-1$$\rangle$$.

Lemma 7.31  

$$\pi_{1}^{}$$(T²#RP²) $$\cong$$ $$\langle$$a, b, c  |  a^-1b^-1ab = c²$$\rangle$$.