(May 2013)
Explain in detail how the Seifert–van Kampen theorem may be used to calculate the fundamental group of a wedge sum \(X\vee Y\) of two spaces under suitable assumptions on the spaces. Clarify what assumptions on the spaces you are using and how you are using them.
Describe the presentation complex \(X_G\) of the group \(G=\langle a,b,c:a^2=1\rangle\) as a wedge sum of familiar spaces. Explain carefully what results you are using.
Solution
Assume \(X\) and \(Y\) are path connected CW complexes, wedged at \(0\)-cells \(x_0,y_0\). Small open neighborhoods of the wedge point can be chosen so that there are open sets \(U,V\subset X\vee Y\) with \[ U\simeq X, \qquad V\simeq Y, \qquad U\cap V\simeq\{*\}. \] Seifert–van Kampen then gives \[ \pi_1(X\vee Y) \cong \pi_1(X)*\pi_1(Y), \] since the amalgamated subgroup \(\pi_1(U\cap V)\) is trivial. More generally, the same argument works whenever the wedge point has neighborhoods allowing such a van Kampen cover.
The presentation complex of \[ G=\langle a,b,c\mid a^2=1\rangle \] has one vertex, three \(1\)-cells \(a,b,c\), and one \(2\)-cell attached along the degree-two loop \(a^2\). The subcomplex consisting of the vertex, the \(a\)-edge, and this \(2\)-cell is the standard CW structure on \(\mathbb{RP}^2\). The \(b\)- and \(c\)-edges are two additional circle summands attached only at the vertex. Hence \[ X_G\cong\mathbb{RP}^2\vee S^1\vee S^1. \] By van Kampen, \[ \pi_1(X_G)\cong(\mathbb Z/2)*\mathbb Z*\mathbb Z, \] which is precisely the presented group.