Find a \(\Delta\)-complex structure on the space \(X\) obtained by identifying three distinct points \(a,b,c\) in the 2-sphere to a point, and compute the simplicial homology groups of \(X\).
Solution
A homeomorphism of \(S^2\) carries \(a,b,c\) to three vertices of the boundary of a tetrahedron; number the vertices \(0=a\), \(1=b\), \(2=c\), \(3=d\). Orient each edge from the lower- to the higher-numbered vertex and write \(x=e_{01}\), \(y=e_{12}\), \(z=e_{02}\), \(p=e_{03}\), \(q=e_{13}\), \(r=e_{23}\), with faces \(f_{012},f_{013},f_{023},f_{123}\).
1. \(X\) is a \(\Delta\)-complex with vertices \(v\) (the image of \(0,1,2\)) and \(d\), the six edges \(x,y,z,p,q,r\), and the four faces.
The identification affects only vertices, so each open edge and open face is still embedded, and the vertex orders are inherited from the tetrahedron.
2. The boundary maps are \(\partial_1x=\partial_1y=\partial_1z=0\), \(\partial_1p=\partial_1q=\partial_1r=d-v\), and\[\partial_2f_{012}=y-z+x,\quad \partial_2f_{013}=q-p+x,\quad \partial_2f_{023}=r-p+z,\quad \partial_2f_{123}=r-q+y.\]
The edges \(x,y,z\) have both endpoints at \(v\), and \(p,q,r\) run from \(v\) to \(d\). For \(i<j<k\), \(\partial[i,j,k]=[j,k]-[i,k]+[i,j]\).
3. \(H_0(X)\cong\ZZ\) and \(\ker\partial_1\) is free on \(x,y,z,q-p,r-p\).
\(\operatorname{im}\partial_1=\ZZ(d-v)\), so \(H_0(X)=(\ZZ v\oplus\ZZ d)/\langle d-v\rangle\cong\ZZ\). The chain \(\alpha x+\beta y+\gamma z+\lambda p+\mu q+\nu r\) has boundary \((\lambda+\mu+\nu)(d-v)\), so it is a cycle exactly when \(\lambda+\mu+\nu=0\), and then \(\lambda p+\mu q+\nu r=\mu(q-p)+\nu(r-p)\).
4. \(H_1(X)\cong\ZZ^2\).
In the basis of step 3, the first three boundaries of step 2 give the relations \(z=x+y\), \(q-p=-x\), and \(r-p=-(x+y)\). The fourth, \((r-p)-(q-p)+y\), then equals \(0\) and adds no relation. So \(\ker\partial_1/\operatorname{im}\partial_2\) is free on the classes of \(x\) and \(y\).
5. \(H_2(X)\cong\ZZ\).
There are no \(3\)-simplices, so \(H_2(X)=\ker\partial_2\). For \(Af_{012}+Bf_{013}+Cf_{023}+Df_{123}\), the coefficients of \(x\), \(z\), \(y\) in its boundary give \(B=-A\), \(C=A\), \(D=-A\), and then the coefficients of \(p,q,r\) vanish. So \(\ker\partial_2=\ZZ(f_{012}-f_{013}+f_{023}-f_{123})\).
6. Q.E.D.
Step 1 gives the \(\Delta\)-complex structure, and steps 3, 4, 5 give \[H_k(X;\ZZ)\cong\begin{cases}\ZZ,&k=0,\\\ZZ^2,&k=1,\\\ZZ,&k=2,\\0,&k\ge3.\end{cases}\]