Find a \(\Delta\)-complex structure on the space \(X\) obtained by identifying the boundaries of three copies of the unit disk together (using identity maps), and compute the simplicial homology groups of \(X\).
Solution
Write the three disks as \(D_1,D_2,D_3\).
1. \(X\) is a \(\Delta\)-complex with vertices \(v,w_1,w_2,w_3\), edges \(e\) (a loop at \(v\)) and \(r_i\colon v\to w_i\), and \(2\)-simplices \(\sigma_1,\sigma_2,\sigma_3\), where \(\sigma_i=[v_0,v_1,v_2]\) has face \([v_0,v_1]\) equal to \(e\) and faces \([v_0,v_2]\), \([v_1,v_2]\) both equal to \(r_i\).
Give the common boundary circle the \(\Delta\)-complex structure with one vertex \(v\) and one loop \(e\). Map \(\sigma_i\) to \(D_i\) with \(v_0,v_1\mapsto v\) and \(v_2\mapsto w_i\), the base onto \(e\), and both remaining faces onto the radial edge \(r_i\). Identifying the two radial faces of a triangle gives the cone on the circle \(e\), a disk, so each \(D_i\) receives this structure. The three structures agree on the boundary circle, which the gluing identifies by the identity, so they form a \(\Delta\)-complex structure on \(X\).
2. \(\partial_2\sigma_i=e\), \(\partial_1r_i=w_i-v\), and \(\partial_1e=0\).
\(\partial_2[v_0,v_1,v_2]=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]=r_i-r_i+e\). \(r_i\) runs from \(v\) to \(w_i\), and \(e\) is a loop at \(v\).
3. Q.E.D.
By step 2:
\(H_2(X)=\ker\partial_2=\{\sum a_i\sigma_i:a_1+a_2+a_3=0\}\), free on \(\sigma_1-\sigma_3\) and \(\sigma_2-\sigma_3\), since there are no \(3\)-simplices.
\(\partial_1(ae+\sum b_ir_i)=\sum b_i(w_i-v)\) vanishes exactly when all \(b_i=0\), so \(\ker\partial_1=\ZZ e=\operatorname{im}\partial_2\) and \(H_1(X)=0\).
\(\operatorname{im}\partial_1\) is generated by \(w_i-v\), so \(H_0(X)=C_0/\operatorname{im}\partial_1\cong\ZZ\).
Hence \[H_n(X)\cong\begin{cases}\ZZ,&n=0,\\0,&n=1,\\\ZZ^2,&n=2,\\0,&n\ge3.\end{cases}\]