Find spaces whose fundamental groups are isomorphic to the following groups. (Here \(\mathbb{Z}/n\) denotes the additive group of integers modulo \(n\).)
\(\mathbb{Z}/n \times \mathbb{Z}/m\)
\(\mathbb{Z}/n_1 \times \mathbb{Z}/n_2 \times \cdots \times \mathbb{Z}/n_k\)
\(\mathbb{Z}/n * \mathbb{Z}/m\). (See E-5DDIL.)
\(\mathbb{Z}/n_1 * \mathbb{Z}/n_2 * \cdots * \mathbb{Z}/n_k\)
Solution
For \(n\ge1\), let \(M_n\) be the space obtained from a circle by attaching a 2-cell by a degree-\(n\) map. The adjoining-a-two-cell theorem gives \[ \pi_1(M_n)\cong\langle a\mid a^n=1\rangle\cong\mathbb Z/n. \]
Take \[ X=M_n\times M_m. \] For path-connected spaces, \(\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)\), hence \[ \pi_1(X)\cong\mathbb Z/n\times\mathbb Z/m. \]
More generally, take \[ X=M_{n_1}\times\cdots\times M_{n_k}. \] Then \[ \pi_1(X)\cong\mathbb Z/n_1\times\cdots\times\mathbb Z/n_k. \]
Take the wedge \[ X=M_n\vee M_m \] at ordinary CW basepoints. The wedge theorem gives \[ \pi_1(X)\cong\pi_1(M_n)*\pi_1(M_m) \cong \mathbb Z/n * \mathbb Z/m. \]
Likewise, \[ X=M_{n_1}\vee\cdots\vee M_{n_k} \] has \[ \pi_1(X)\cong \mathbb Z/n_1*\cdots*\mathbb Z/n_k. \] These spaces are finite two-dimensional CW complexes.