Recalling that \(S^2=\{(x,y,z)\in\RR^3:x^2+y^2+z^2=1\}\), define the equator of \(S^2\) to be \(\{(x,y,z)\in S^2:z=0\}\), so that the equator is homeomorphic to \(S^1\). Let \(Z_1\) and \(Z_2\) be disjoint copies of the \(2\)-sphere \(S^2\), let \(f\) be a homeomorphism from the equator of \(Z_1\) to the equator of \(Z_2\), and let \(Z\) be the quotient space obtained from \(Z_1\sqcup Z_2\) by identifying the equator of \(Z_1\) to the equator of \(Z_2\) via \(f\). Find a \(\Delta\)-complex structure on the space \(Z\), and compute the simplicial homology groups of \(Z\).
Solution
1. \(Z\) has a \(\Delta\)-complex structure with \(7\) vertices, \(15\) edges, and \(12\) triangles: triangulate the equator of \(Z_1\) by three vertices and three edges, transport this triangulation to the equator of \(Z_2\) by \(f\), and cone the equatorial triangle to one interior vertex in each of the four hemispheres.
The two sphere triangulations agree on the identified equator by construction, so the quotient is a simplicial complex with \(3+4=7\) vertices, \(3+4\cdot3=15\) edges, and \(4\cdot3=12\) triangles.
2. \(H_2^\Delta(Z)\cong\ZZ^3\).
Let \(E\) be the oriented equatorial \(1\)-cycle and \(C_1,\ldots,C_4\) the sums of the three triangles in the four hemispheres, oriented so that \(\partial C_1=E\), \(\partial C_2=-E\), \(\partial C_3=E\), \(\partial C_4=-E\); within each hemisphere the radial edges cancel. In a \(2\)-cycle, cancellation along the radial edges forces the three coefficients in each hemisphere to be equal, so every \(2\)-cycle is \(\sum r_iC_i\), with boundary \((r_1-r_2+r_3-r_4)E\). The kernel of this primitive linear form on \(\ZZ^4\) is free of rank \(3\), and there are no \(3\)-simplices.
3. \(\pi_1(Z)=1\).
Apply the Seifert–van Kampen theorem to open neighborhoods of \(Z_1\) and \(Z_2\) in \(Z\) that deformation retract onto them and whose intersection deformation retracts onto the equator. \(\pi_1(Z)\) is the pushout of \(\pi_1(S^2)=1\leftarrow\pi_1(S^1)\to\pi_1(S^2)=1\), which is trivial.
4. Q.E.D.
Simplicial and singular homology agree for a \(\Delta\)-complex. \(Z\) is connected, so \(H_0^\Delta(Z)\cong\ZZ\); by step 3, \(H_1^\Delta(Z)\), the abelianization of \(\pi_1(Z)\), is \(0\); and with step 2 and no simplices above dimension \(2\), \[H_n^\Delta(Z)\cong\begin{cases}\ZZ,&n=0,\\\ZZ^3,&n=2,\\0,&\text{otherwise}.\end{cases}\]