Give a \(\Delta\)-complex structure for a standard 2-simplex \(\Delta\) and write down its chain complex. Explicitly describe the boundary homomorphisms.
Solution
Let the vertices of the standard \(2\)-simplex be \(v_0,v_1,v_2\), orient the edges by increasing indices, and let \[ e_{01}=[v_0,v_1],\quad e_{02}=[v_0,v_2],\quad e_{12}=[v_1,v_2], \] with oriented \(2\)-simplex \(\sigma=[v_0,v_1,v_2]\). This is the standard \(\Delta\)-complex structure.
The chain complex is \[ 0\to\mathbb Z\langle\sigma\rangle \xrightarrow{\partial_2} \mathbb Z\langle e_{01},e_{02},e_{12}\rangle \xrightarrow{\partial_1} \mathbb Z\langle v_0,v_1,v_2\rangle \to0, \] where \[ \partial_2\sigma=e_{12}-e_{02}+e_{01} \] and \[ \partial_1e_{01}=v_1-v_0, \qquad \partial_1e_{02}=v_2-v_0, \qquad \partial_1e_{12}=v_2-v_1. \] These satisfy \(\partial_1\partial_2=0\), as required.