Compute the groups \(H_i(\mathbb{RP}^m \times \mathbb{RP}^n; G)\) and \(H^i(\mathbb{RP}^m \times \mathbb{RP}^n; G)\) for \(G = \mathbb{Z}\) and \(\mathbb{Z}_2\) via the cellular chain and cochain complexes. [See Example 3B.4.]
Solution
Give \(\mathbb{RP}^r\) its standard CW structure with one cell \(e_j\) in each dimension \(0\le j\le r\). Its cellular chain complex over \(\mathbb Z\) has \[ \partial e_j=\begin{cases} 0,&j\text{ odd},\\ 2e_{j-1},&j\text{ even}. \end{cases} \] The product CW structure on \(\mathbb{RP}^m\times\mathbb{RP}^n\) has one cell \(e_p\times e_q\) for each \(0\le p\le m\), \(0\le q\le n\), and its cellular chain complex is the tensor product of the two cellular complexes, with \[ \partial(e_p\otimes e_q) =(\partial e_p)\otimes e_q+(-1)^p e_p\otimes(\partial e_q). \]
For a compact notation define \[ F_r=\{0\}\cup\bigl(\{r\}\text{ if $r$ is odd}\bigr), \qquad T_r=\{j:1\le j<r,\ j\text{ odd}\}. \] Thus the integral homology of \(\mathbb{RP}^r\) is \(\mathbb Z\) in degrees in \(F_r\), \(\mathbb Z_2\) in degrees in \(T_r\), and zero otherwise.
The cellular tensor complex gives the integral Künneth decomposition \[ H_i(\mathbb{RP}^m\times\mathbb{RP}^n;\mathbb Z) \cong \bigoplus_{p+q=i}H_p(\mathbb{RP}^m)\otimes H_q(\mathbb{RP}^n) \oplus \bigoplus_{p+q=i-1}\operatorname{Tor}(H_p(\mathbb{RP}^m),H_q(\mathbb{RP}^n)). \] Since the only torsion is \(\mathbb Z_2\), this may be written explicitly as \[ H_i\cong \mathbb Z^{f_i}\oplus(\mathbb Z_2)^{t_i}, \] where \[ f_i=\#\{(p,q)\in F_m\times F_n:p+q=i\}, \] and \[ \begin{aligned} t_i={}&\#\{(p,q):p+q=i,\ (p,q)\in (T_m\times F_n)\cup(F_m\times T_n)\cup(T_m\times T_n)\}\\ &+\#\{(p,q)\in T_m\times T_n:p+q+1=i\}. \end{aligned} \] This is the complete integral homology calculation.
The integral cohomology follows from the universal coefficient theorem. Since \(\operatorname{Hom}(\mathbb Z_2,\mathbb Z)=0\) and \(\operatorname{Ext}(\mathbb Z_2,\mathbb Z)\cong\mathbb Z_2\), \[ \boxed{H^i(\mathbb{RP}^m\times\mathbb{RP}^n;\mathbb Z) \cong \mathbb Z^{f_i}\oplus(\mathbb Z_2)^{t_{i-1}}.} \]
With \(\mathbb Z_2\) coefficients every cellular differential is zero, since each nonzero integral differential is multiplication by \(2\). Hence both cellular homology and cellular cohomology have one \(\mathbb Z_2\) basis vector for each product cell of total dimension \(i\). Therefore \[ \boxed{H_i(-;\mathbb Z_2)\cong H^i(-;\mathbb Z_2) \cong(\mathbb Z_2)^{N_i},} \] where \[ N_i=\#\{(p,q):0\le p\le m,\ 0\le q\le n,\ p+q=i\}. \] Equivalently, \[ N_i=\max\bigl(0,\min(m,i)-\max(0,i-n)+1\bigr). \]