Compute \(H_*(\RP^2\cross \RP^3; \ZZ)\)
Solution
1. The integral homology of the factors is\[ H_i(\mathbb{RP}^2)= \begin{cases} \mathbb Z,&i=0,\\ \mathbb Z/2,&i=1,\\ 0,&\text{otherwise}, \end{cases} \]and\[ H_i(\mathbb{RP}^3)= \begin{cases} \mathbb Z,&i=0,3,\\ \mathbb Z/2,&i=1,\\ 0,&\text{otherwise}. \end{cases} \]
These are the standard cellular homology groups of real projective spaces.
2. The Künneth theorem gives\[ \boxed{H_n(\mathbb{RP}^2\times\mathbb{RP}^3;\mathbb Z)\cong \begin{cases} \mathbb Z,&n=0,\\ (\mathbb Z/2)^2,&n=1,\\ \mathbb Z/2,&n=2,\\ \mathbb Z\oplus\mathbb Z/2,&n=3,\\ \mathbb Z/2,&n=4,\\ 0,&n\ge5. \end{cases}} \]
The tensor terms give \[ \begin{array}{c|ccccc} n&0&1&2&3&4\\ \hline \bigoplus_{i+j=n}H_i(\mathbb{RP}^2)\otimes H_j(\mathbb{RP}^3) &\mathbb Z&(\mathbb Z/2)^2&\mathbb Z/2&\mathbb Z&\mathbb Z/2. \end{array} \] The only nonzero Tor contribution is \[ \operatorname{Tor}(H_1(\mathbb{RP}^2),H_1(\mathbb{RP}^3)) \cong\operatorname{Tor}(\mathbb Z/2,\mathbb Z/2) \cong\mathbb Z/2, \] which appears in degree \(3\). The Künneth short exact sequence splits noncanonically, giving the displayed groups.