1.2. Van Kampen’s Theorem#
The van Kampen theorem gives a method for computing the fundamental groups of spaces that can be decomposed into simpler spaces whose fundamental groups are already known. By systematic use of this theorem one can compute the fundamental groups of a very large number of spaces. We shall see for example that for every group \(G\) there is a space \(X_G\) whose fundamental group is isomorphic to \(G\).
To give some idea of how one might hope to compute fundamental groups by decomposing spaces into simpler pieces, let us look at an example. Consider the space \(X\) formed by two circles \(A\) and \(B\) intersecting in a single point, which we choose as the basepoint \(x_0\). By our preceding calculations we know that \(\pi_1(A)\) is infinite cyclic, generated by a loop \(a\) that goes once around \(A\).
Similarly, \(\pi_1(B)\) is a copy of \(\mathbb{Z}\) generated by a loop \(b\) going once around \(B\). Each product of powers of \(a\) and \(b\) then gives an element of \(\pi_1(X)\). For example, the product \(a^5b^2a^{-3}ba^2\) is the loop that goes five times around \(A\), then twice around \(B\), then three times around \(A\) in the opposite direction, then once around \(B\), then twice around \(A\). The set of all words like this consisting of powers of \(a\) alternating with powers of \(b\) forms a group usually denoted \(\mathbb{Z}*\mathbb{Z}\). Multiplication in this group is defined just as one would expect, for example \((b^4 a^5 b^2 a^{-3})(a^4 b^{-1}ab^3) = b^4a^5b^2ab^{-1}ab^3\). The identity element is the empty word, and inverses are what they have to be, for example \((ab^2a^{-3}b^{-4})^{-1} = b^4a^3b^{-2}a^{-1}\). It would be very nice if such words in \(a\) and \(b\) corresponded exactly to elements of \(\pi_1(X)\), so that \(\pi_1(X)\) was isomorphic to the group \(\mathbb{Z}*\mathbb{Z}\). The van Kampen theorem will imply that this is indeed the case.
Similarly, if \(X\) is the union of three circles touching at a single point, the van Kampen theorem will imply that \(\pi_1(X)\) is \(\mathbb{Z}*\mathbb{Z}*\mathbb{Z}\), the group consisting of words in powers of three letters \(a,\,b,\,c\). The generalization to a union of any number of circles touching at one point will also follow.
The group \(\mathbb{Z} * \mathbb{Z}\) is an example of a general construction called the free product of groups. The statement of van Kampen’s theorem will be in terms of free products, so before stating the theorem we will make an algebraic digression to describe the construction of free products in some detail.