Let \(X_n\) be a space whose homology groups are given by \(H_k(X_n; \mathbb{Z}) \cong \mathbb{Z}/k\mathbb{Z}\) for \(0 \leq k \leq n\) and which vanish for \(k > n\). Compute the homology \(H_*(X_3 \times X_5; \mathbb{Z})\).
Solution
1. Interpreting \(\mathbb Z/0\mathbb Z\) as \(\mathbb Z\) and \(\mathbb Z/1\mathbb Z\) as \(0\), the nonzero homology groups are\[ H_0(X_3)=\mathbb Z,\quad H_2(X_3)=\mathbb Z/2,\quad H_3(X_3)=\mathbb Z/3, \]and\[ H_0(X_5)=\mathbb Z,\ H_2=\mathbb Z/2,\ H_3=\mathbb Z/3,\ H_4=\mathbb Z/4,\ H_5=\mathbb Z/5. \]
This is the given formula \(H_k(X_n)=\mathbb Z/k\mathbb Z\) in the indicated range.
2. The tensor terms in the integral Künneth theorem give\[ \begin{array}{c|c} q&\displaystyle\bigoplus_{i+j=q}H_i(X_3)\otimes H_j(X_5)\\ \hline 0&\mathbb Z\\ 2&(\mathbb Z/2)^2\\ 3&(\mathbb Z/3)^2\\ 4&\mathbb Z/4\oplus\mathbb Z/2\\ 5&\mathbb Z/5\\ 6&\mathbb Z/2\oplus\mathbb Z/3\\ \end{array} \]with zero tensor terms in the other degrees.
Use \(\mathbb Z/a\otimes\mathbb Z/b\cong\mathbb Z/\gcd(a,b)\) for \(a,b>1\) and tensoring with \(\mathbb Z\) unchanged.
3. The only nonzero Tor terms are\[ \operatorname{Tor}(\mathbb Z/2,\mathbb Z/2)\cong\mathbb Z/2 \]in total degree \(5\), and\[ \operatorname{Tor}(\mathbb Z/2,\mathbb Z/4)\oplus \operatorname{Tor}(\mathbb Z/3,\mathbb Z/3) \cong\mathbb Z/2\oplus\mathbb Z/3 \]in total degree \(7\).
A Tor term from bidegree \((i,j)\) contributes to \(H_{i+j+1}\), and \(\operatorname{Tor}(\mathbb Z/a,\mathbb Z/b)\cong\mathbb Z/\gcd(a,b)\).
4. Therefore, using the noncanonical splitting in the Künneth theorem,\[ \boxed{H_q(X_3\times X_5;\mathbb Z)\cong \begin{cases} \mathbb Z,&q=0,\\ (\mathbb Z/2)^2,&q=2,\\ (\mathbb Z/3)^2,&q=3,\\ \mathbb Z/4\oplus\mathbb Z/2,&q=4,\\ \mathbb Z/10,&q=5,\\ \mathbb Z/6,&q=6,7,\\ 0,&\text{otherwise}. \end{cases}} \]
In degree \(5\), \(\mathbb Z/5\oplus\mathbb Z/2\cong\mathbb Z/10\); in degrees \(6\) and \(7\), \(\mathbb Z/2\oplus\mathbb Z/3\cong\mathbb Z/6\).