(Jan ’12) Let \(X\) be a space obtained by identifying the three vertices of a standard 2-simplex.
Describe a structure of a \(\Delta\)-complex on \(X\).
Write down the chain complex corresponding to the \(\Delta\)-complex in (a).
Compute the homology of \(X\).
Solution
Let the vertices of the standard \(2\)-simplex be \(v_0,v_1,v_2\), and identify all three to a single vertex \(v\). The three edges descend to three distinct \(1\)-simplices. Orient them by \[ a=[v_0,v_1],\qquad b=[v_1,v_2],\qquad c=[v_0,v_2]. \] The interior of the original triangle is one \(2\)-simplex \(\sigma=[v_0,v_1,v_2]\). This gives the desired \(\Delta\)-complex structure.
The simplicial chain groups are \[ 0\longrightarrow C_2\cong\mathbb Z \xrightarrow{\partial_2} C_1\cong\mathbb Z^3 \xrightarrow{\partial_1} C_0\cong\mathbb Z \longrightarrow0. \] Since every edge begins and ends at \(v\), \(\partial_1=0\). The oriented boundary of \(\sigma\) is \[ \partial_2\sigma=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]=a+b-c. \] Thus, in the indicated bases, \[ \partial_2(1)=(1,1,-1), \qquad \partial_1=0. \] The vector \((1,1,-1)\) is primitive and nonzero, so \(\partial_2\) is injective and its cokernel is free of rank \(2\). Hence \[ H_n(X;\mathbb Z)\cong \begin{cases} \mathbb Z,&n=0,\\ \mathbb Z^2,&n=1,\\ 0,&n\ge2. \end{cases} \]