Find the (simplicial) homology groups of the space \(X\) obtained from the 3-simplex \(\Delta^3=[v_0,v_1,v_2,v_3]\) by gluing the face \([v_0,v_1,v_2]\) to the face \([v_0,v_2,v_3]\) and the face \([v_0,v_1,v_3]\) to the face \([v_1,v_2,v_3]\) (respecting the ordering of vertices as written, i.e., the first gluing map sends \(v_1\) to \(v_2\), and so on).
Solution
1. \(X\) is a \(\Delta\)-complex with one vertex \(v\), one edge \(e\), two \(2\)-simplices \(a=[v_0,v_1,v_2]=[v_0,v_2,v_3]\) and \(b=[v_0,v_1,v_3]=[v_1,v_2,v_3]\), and one \(3\)-simplex \(\sigma=[v_0,v_1,v_2,v_3]\).
The first gluing sends \(v_0\mapsto v_0\), \(v_1\mapsto v_2\), \(v_2\mapsto v_3\), so \(v_1\sim v_2\sim v_3\), and on edges \([v_0,v_1]\sim[v_0,v_2]\), \([v_1,v_2]\sim[v_2,v_3]\), \([v_0,v_2]\sim[v_0,v_3]\). The second sends \(v_0\mapsto v_1\), \(v_1\mapsto v_2\), \(v_3\mapsto v_3\), so \(v_0\sim v_1\), and on edges \([v_0,v_1]\sim[v_1,v_2]\), \([v_0,v_3]\sim[v_1,v_3]\), \([v_1,v_3]\sim[v_2,v_3]\). Chaining these, all four vertices are identified, and \([v_0,v_1]\sim[v_0,v_2]\sim[v_1,v_2]\sim[v_2,v_3]\sim[v_1,v_3]\sim[v_0,v_3]\).
2. The simplicial chain complex is \(0\to\ZZ\sigma\xrightarrow{d_3}\ZZ a\oplus\ZZ b\xrightarrow{d_2}\ZZ e\xrightarrow{d_1}\ZZ v\to0\) with \(d_3(\sigma)=2b-2a\), \(d_2(a)=d_2(b)=e\), and \(d_1=0\).
By step 1, \(d_3\sigma=[v_1,v_2,v_3]-[v_0,v_2,v_3]+[v_0,v_1,v_3]-[v_0,v_1,v_2]=b-a+b-a\); \(d_2[v_0,v_1,v_2]=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]=e-e+e=e\), and likewise \(d_2[v_0,v_1,v_3]=e\); and \(d_1e=v-v=0\).
3. Q.E.D.
By step 2: \(H_0(X)=\ZZ v\cong\ZZ\); \(H_1(X)=\ZZ e/\ZZ e=0\); \(\ker d_2=\{ma+nb:m+n=0\}=\ZZ(a-b)\) and \(\operatorname{im}d_3=\ZZ\cdot2(a-b)\), so \(H_2(X)\cong\ZZ/2\); and \(d_3\) is injective, so \(H_3(X)=0\). Hence \(H_0\cong\ZZ\), \(H_1=0\), \(H_2\cong\ZZ/2\), \(H_3=0\), and \(H_k=0\) for \(k\ge4\).