(May 2013) Let \(Y\) be the standard \(3\)-simplex \(\Delta^3\) with a total ordering on its four vertices. Let \(X\) be the \(\Delta\)-complex obtained from \(Y\) by identifying, for each \(k\leq 3\), all of its \(k\)-dimensional faces such that the identifications respect the vertex ordering. Thus \(X\) has a single \(k\)-simplex for each \(k\leq 3\). Compute the simplicial homology groups of the \(\Delta\)-complex \(X\).
Solution
Let \(e_k\) be the unique \(k\)-simplex of \(X\) for \(0\le k\le 3\), so \(\Delta_k(X)=\mathbb Z e_k\) for \(0\le k\le3\) and \(\Delta_k(X)=0\) for \(k\ge4\). Every face of \(e_k\) of dimension \(k-1\) is \(e_{k-1}\).
1. \(\partial_1=0\), \(\partial_2(e_2)=e_1\), and \(\partial_3=0\).
\(\partial_1 e_1=[v_1]-[v_0]=e_0-e_0=0\). \(\partial_2 e_2=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]=e_1-e_1+e_1=e_1\). \(\partial_3 e_3=[v_1,v_2,v_3]-[v_0,v_2,v_3]+[v_0,v_1,v_3]-[v_0,v_1,v_2]=e_2-e_2+e_2-e_2=0\).
2. Q.E.D.
By step 1, \(\partial_2\colon\mathbb Z\to\mathbb Z\) is an isomorphism and \(\partial_1=\partial_3=0\). Hence \(H_0=\mathbb Z e_0/0\cong\mathbb Z\), \(H_1=\mathbb Z e_1/\mathbb Z e_1=0\), \(H_2=\ker\partial_2/\operatorname{im}\partial_3=0\), \(H_3=\ker\partial_3=\mathbb Z e_3\cong\mathbb Z\), and \(H_k=0\) for \(k\ge4\): \[ \boxed{H_k(X)\cong\begin{cases}\mathbb{Z}, & k = 0, 3,\\ 0, & \text{otherwise}.\end{cases}} \]