Given a group \(G = <A : R>\), show how to construct a CW-complex \(X\) such that \(\pi_1(X) = G\).
Solution
Goal: Given a group presentation \(G = \langle A \mid R \rangle\) with generator set \(A\) and relator set \(R\), construct a 2-dimensional CW complex \(X\) such that \(\pi_1(X, x_0) \cong G\).
1. Construct the 0-skeleton \(X^0\) and 1-skeleton \(X^1\).
1.1. Set the 0-skeleton \(X^0 = \{x_0\}\) to be a single vertex.
1.2. For each generator \(a \in A\), attach a 1-cell \(e_a^1 = [0, 1]\) to \(X^0\) by identifying both endpoints \(0, 1 \in \partial e_a^1\) to the vertex \(x_0\).
1.3. The 1-skeleton is the wedge sum of circles indexed by \(A\): \[X^1 = \bigvee_{a \in A} S_a^1.\]
1.4. By the Seifert-van Kampen theorem, the fundamental group of \(X^1\) is the free group on \(A\): \[\pi_1(X^1, x_0) \cong F(A) = \langle A \mid \emptyset \rangle.\]
1.5. The wedge of circles indexed by \(A\) has fundamental group the free product of one copy of \(\mathbb{Z}\) per circle, which is the free group \(F(A)\).
2. Attach 2-cells to form the 2-skeleton \(X = X^2\).
2.1. Each relator \(r \in R\) is a reduced word in the free group \(F(A)\), say \(r = a_{i_1}^{\epsilon_1} a_{i_2}^{\epsilon_2} \cdots a_{i_k}^{\epsilon_k}\) with \(a_{i_j} \in A\) and \(\epsilon_j \in \{\pm 1\}\).
2.2. Define a continuous attaching map \(\phi_r \colon \partial D^2 \cong S^1 \to X^1\) by parameterizing \(S^1 = [0, 1] / (0 \sim 1)\) into \(k\) equal subintervals \([(j-1)/k, j/k]\), and mapping the \(j\)-th subinterval to traverse the loop \(e_{a_{i_j}}^1\) in the positive direction if \(\epsilon_j = 1\) and in the reverse direction if \(\epsilon_j = -1\).
2.3. Under this definition, the loop \(\phi_r\) represents the homotopy class \([\phi_r] = r \in \pi_1(X^1, x_0)\).
2.4. Attach a 2-cell \(e_r^2 \cong D^2\) along \(\phi_r\) for each relator \(r \in R\): \[X = X^2 = X^1 \cup_{\{\phi_r\}_{r \in R}} \left( \bigsqcup_{r \in R} D_r^2 \right).\]
2.5. The attaching map \(\phi_r\) is defined by traversing the 1-cells according to the letters of the word \(r\), so by construction \([\phi_r] = r\) in \(\pi_1(X^1, x_0)\).
3. Apply the cellular Seifert-van Kampen theorem to compute \(\pi_1(X, x_0)\).
3.1. By the standard theorem on fundamental groups of 2-dimensional CW complexes (e.g. Hatcher Proposition 1.26), attaching a family of 2-cells \(\{e_r^2\}_{r \in R}\) along maps \(\phi_r \colon \partial D^2 \to X^1\) induces an isomorphism: \[\pi_1(X, x_0) \cong \pi_1(X^1, x_0) / N,\] where \(N = \langle\langle [\phi_r] : r \in R \rangle\rangle\) is the normal subgroup of \(\pi_1(X^1, x_0)\) generated by the homotopy classes of the attaching maps.
3.2. By 1, \(\pi_1(X^1, x_0) = F(A)\).
3.3. By 2, \([\phi_r] = r \in F(A)\) for each \(r \in R\), so \(N = \langle\langle R \rangle\rangle\).
3.4. Thus, \(\pi_1(X, x_0) \cong F(A) / \langle\langle R \rangle\rangle \cong \langle A \mid R \rangle = G\).
3.5. The cellular van Kampen theorem (<2>1) gives \(\pi_1(X, x_0) \cong F(A)/N\); by 1 and 2, \(F(A) = \pi_1(X^1, x_0)\) and \(N = \langle\langle R \rangle\rangle\), so the quotient is \(\langle A \mid R \rangle = G\).
4. Q.E.D.
4.1. Steps <1>1–<1>3 explicitly construct the standard presentation complex \(X = X(A; R)\) satisfying \(\pi_1(X) \cong G\).