Describe the CW complex structure of the real projective plane \(\mathbb{RP}^2\), compute its cellular boundary maps, and compute its simplicial/cellular homology and cohomology groups with coefficients in \(\mathbb{Z}\) and \(\mathbb{Z}_2\).
Solution
1. The standard CW structure on \(\mathbb{RP}^2\) has one cell \(e^k\) in each dimension \(k=0,1,2\).
1.1. The \(1\)-skeleton is \(\mathbb{RP}^1\cong S^1\).
1.2. The \(2\)-cell is attached by the degree-\(2\) map \(S^1\to S^1\).
2. With integer coefficients, the cellular chain complex is\[ 0\longrightarrow\mathbb Z \xrightarrow{\,2\,}\mathbb Z \xrightarrow{\,0\,}\mathbb Z \longrightarrow0. \]Hence\[ H_k(\mathbb{RP}^2;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=0,\\ \mathbb Z/2,&k=1,\\ 0,&k\ge2. \end{cases} \]
3. With \(\mathbb Z_2\) coefficients, multiplication by \(2\) becomes zero, so the cellular chain complex is\[ 0\longrightarrow\mathbb Z_2 \xrightarrow{0}\mathbb Z_2 \xrightarrow{0}\mathbb Z_2 \longrightarrow0. \]Therefore\[ H_k(\mathbb{RP}^2;\mathbb Z_2)\cong \begin{cases} \mathbb Z_2,&k=0,1,2,\\ 0,&k>2. \end{cases} \]
4. The integral cellular cochain complex is\[ 0\longleftarrow\mathbb Z \xleftarrow{0}\mathbb Z \xleftarrow{2}\mathbb Z \longleftarrow0, \]so\[ H^k(\mathbb{RP}^2;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=0,\\ \mathbb Z/2,&k=2,\\ 0,&\text{otherwise}. \end{cases} \]
5. With \(\mathbb Z_2\) coefficients all cellular coboundaries vanish, hence\[ H^k(\mathbb{RP}^2;\mathbb Z_2)\cong \begin{cases} \mathbb Z_2,&k=0,1,2,\\ 0,&k>2. \end{cases} \]These cellular groups agree with singular homology and cohomology; any triangulation gives the same groups simplicially.